INTEGRALS OF GEOMETRIC RANDOM VARIABLES INVOLVING EIGHT-SHAPED GEODESICS ON HYPERBOLIC SURFACES

Victor Le Guilloux

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Victor Le Guilloux. INTEGRALS OF GEOMETRIC RANDOM VARIABLES INVOLVING EIGHT-SHAPED GEODESICS ON HYPERBOLIC SURFACES. 2025. hal-05048889v2

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Distributed under a Creative Commons Attribution 4.0 International License
INTEGRALS OF GEOMETRIC RANDOM VARIABLES INVOLVING
EIGHT-SHAPED GEODESICS ON HYPERBOLIC SURFACES

VICTOR LE GUILLOUX

ABSTRACT. Given a hyperbolic surface X and any closed geodesic $\gamma$ on X, Mirzakhani provided an asymptotic equivalent of the number $N_{\gamma}(a)(X)$ of geodesics of length at most $a$ in the mapping class group orbit of $\gamma$, when $a \to \infty$. After integrating this result on the moduli space, she obtained a similar asymptotic equivalent of the Weil-Petersson expectation $\mathbb{E}[N_{\gamma}(a)]$. In this paper we use Mirzakhani's integration formula in order to integrate random variables on the moduli space which definition involves lengths of eight-shaped geodesics, and we compute the asymptotic behavior of the density of these geometric random variables by using tools developed by Anantharaman and Monk. As a consequence, we improve the asymptotic expansion of the average counting function $a \mapsto \mathbb{E}[N_{\gamma}(a)]$ for geodesics $\gamma$ with exactly one self-intersection and we provide an explicit computation of the leading term of this expansion when $\gamma$ is an eight-shaped geodesic on a $n$-holed sphere.

CONTENTS

<table><tr><td>1. Introduction</td><td>1</td></tr><tr><td>2. An integration formula for eight-shaped geodesics</td><td>6</td></tr><tr><td>3. Pseudo-convolution of Friedman-Ramanujan functions</td><td>11</td></tr><tr><td>4. Application to the integration of geometric random variables</td><td>19</td></tr><tr><td>5. Conclusion</td><td>27</td></tr><tr><td>References</td><td>27</td></tr></table>

1. INTRODUCTION

1.1. Main results. We fix two integers $g, n \ge 0$ and a connected and oriented surface $S_{g,n}$ of genus $g$ with $n$ labelled boundaries $\alpha_1, \dots, \alpha_n$, and we assume that $2g - 2 + n > 0$. This enables us to consider the moduli space $M_{g,n}(L)$ of hyperbolic metrics on $S_{g,n}$ with geodesic boundaries of lengths given by $L = (L_1, \dots, L_n) \in \mathbb{R}^n_{\ge 0}$ (with the convention that a boundary of length 0 is a cusp, in which case the hyperbolic surface is non compact but has a finite volume). In this article we study random variables defined over $M_{g,n}(L)$ involving lengths of closed geodesics with exactly one self-intersection, in a way that we will explain below. Our probabilistic framework is the Weil-Petersson model, i.e. we endow the moduli space $M_{g,n}(L)$ with the Weil-Petersson measure $\mu_{WP}$, we denote by $V_{g,n}(L)$ the Weil-Petersson volume of $M_{g,n}(L)$, and we denote by $\mathbb{E}_{g,n}^L$ the expectation for the probability measure $\frac{1}{\sqrt{V_{g,n}(L)}}\mu_{WP}$.

Let $\gamma$ be a curve on $S_{g,n}$. We denote by $O_\gamma$ the mapping class group orbit of the free homotopy class $[\gamma]$ of $\gamma$. Given a hyperbolic metric $X \in M_{g,n}(L)$, any orientation-preserving homeomorphism $f : S_{g,n} \to X$ and any test function $F : \mathbb{R} \to \mathbb{R}$ (e.g. compactly supported), we denote:

$$F^{\gamma}(X) = \sum_{[\gamma'] \in O_{\gamma}} F(\ell_{f(\gamma')}(X)), \qquad (1)$$

1
where $\ell_{f(\gamma')}(X)$ is the length of the unique geodesic of $X$ freely homotopic to $f(\gamma')$. Here $F^{\gamma}(X)$ does not depend on the choice of the homeomorphism $f$, hence $F^{\gamma}$ is a well-defined random variable on $M_{g,n}(L)$. Our main result is the following.

**Theorem 1.1.** Let $\gamma$ be a closed curve on $S_{g,n}$ with exactly one self-intersection. There exists a locally integrable function $V_{\mathcal{O}_{\gamma}}(\cdot, L) : \mathbf{R} \to \mathbf{R}_{\ge 0}$, depending only on $\mathcal{O}_{\gamma}$, such that for every test function $F : \mathbf{R} \to \mathbf{R}$, if $F^{\gamma}$ is defined as (1), we have:

$$ \mathbb{E}_{g,n}^{L}[F^{\gamma}] = \int_{\mathbf{R}} F(\ell) V_{\mathcal{O}_{\gamma}}(\ell, L) \frac{d\ell}{V_{g,n}(L)}. $$

Moreover, there exist a polynomial $P_{\gamma}^{L}$ of degree $6g - 7 + 2n$ and constants $c_0, c > 0$ such that for every $m \ge 1$:

$$ \int_{0}^{m} |V_{\mathcal{O}_{\gamma}}(\ell, L) - P_{\gamma}^{L}(\ell)| e^{\ell} d\ell \le c_0 (1+m)^{c} e^{\frac{m}{2}}. \quad (2) $$

As a consequence we obtain the following result.

**Corollary 1.2.** Let $\gamma$ be a closed curve on $S_{g,n}$ with exactly one self-intersection and $a \ge 0$. Given a hyperbolic metric $X \in M_{g,n}(L)$, we denote by $N_{\gamma}(a)(X)$ the number of geodesics on $X$ belonging to the same mapping class group orbit as $\gamma$, with length at most $a$. There exist a polynomial $Q_{\gamma}^{L}$ of degree $6g - 6 + 2n$ and a constant $c > 0$ such that:

$$ \mathbb{E}_{g,n}^{L}[N_{\gamma}(a)] = Q_{\gamma}^{L}(a) + \mathcal{O}_{a \to \infty}((1+a)^{c}e^{-\frac{a}{2}}). $$

We even provide in Paragraphs 4.1.2 and 4.2.3 examples of orbits of curves where the exponential remainder term is non-zero.

In the rest of this article, we will work with fixed values of $g, n$ and $L$. Hence from now on we will only denote by $\mathbb{E}$ the Weil-Petersson expectation on $M_{g,n}(L)$.

**1.2. The moduli space and the theory of random hyperbolic surfaces.** We denote by $\text{Diff}(S_{g,n})$ the group of orientation-preserving diffeomorphisms $\varphi$ of $S_{g,n}$ such that $\varphi(\alpha_i) = \alpha_i$ for all $1 \le i \le n$ and by $\text{Diff}_0(S_{g,n})$ the subgroup of $\text{Diff}(S_{g,n})$ of diffeomorphisms isotopic to the identity. The mapping class group is defined by:

$$ \text{MCG}_{g,n} = \text{Diff}(S_{g,n})/\text{Diff}_0(S_{g,n}). $$

This group acts on the Teichmüller space $\mathcal{T}_{g,n}(L)$, which is the set of equivalence classes of marked hyperbolic surfaces homeomorphic to $S_{g,n}$. A marked hyperbolic surface is a couple $(X, f)$, where $X$ is an oriented hyperbolic surface with labelled geodesic boundaries $\beta_1, \dots, \beta_n$ of lengths $L_1, \dots, L_n$ respectively, and $f: S_{g,n} \to X$ is a homeomorphism which preserves the orientation of $X$ and the labelling of the boundaries. Two marked surfaces $(X, f)$ and $(Y, g)$ are equivalent if $g \circ f^{-1}$ is isotopic to an isometry. The moduli space is the quotient space:

$$ \mathcal{M}_{g,n}(L) = \mathcal{T}_{g,n}(L)/\text{MCG}_{g,n}. $$

This space is naturally equipped with the Weil-Petersson measure $\mu_{WP}$. The moduli space $M_{g,n}(L)$ has a finite volume $V_{g,n}(L)$ and Mirzakhani proved the following major result:

**Theorem 1.3 ([Mir07a]).** The Weil-Petersson volume $V_{g,n}(L)$ is a polynomial function in the variables $L_1, \dots, L_n$. More precisely we can write:

$$ V_{g,n}(L) = \sum_{i_1+\cdots+i_n \le 3g-3+n} C_{i_1 \cdots i_n} L_1^{2i_1} \cdots L_n^{2i_n}, $$

where $C_{i_1 \cdots i_n} \in \pi^{6g-6+2n-2(i_1+\cdots+i_n)} \mathbb{Q}_{>0}$.
Mirzakhani even developed in [Mir07b] a topological recursion based on her integration formula (Theorem 1.4) and McShane Identities to compute explicitly the volume polynomials $V_{g,n}(L)$.

After renormalization of $\mu_{WP}$ so its total mass is 1, $M_{g,n}(L)$ therefore becomes a probability space which is our framework for the study of random hyperbolic surfaces.

### 1.3. Mirzakhani's integration formula and other results on geodesic counting.
In addition to Theorem 1.3, Mirzakhani developed an integration formula which makes possible to compute the expectation of a certain class of random variables defined on the moduli space. Let us describe these random variables.

In what follows, all the curves we consider are oriented.

A closed curve $\gamma$ on a surface $X$ is said to be in minimal position if it is immersed, if all its self-intersecting points are double points and if $\gamma$ has the minimal amount of self-intersections among the immersed curves homotopic to $\gamma$ with at most double points. The curve $\gamma$ is said to be essential if it is non-contractible and not homotopic to any of the boundaries and cusps of $X$. Each closed essential curve on a hyperbolic surface is freely homotopic to exactly one closed geodesic. If $\gamma$ is an essential curve on $X$, we denote by $\ell_{\gamma}(X)$ the length of the unique geodesic freely homotopic to $\gamma$ (hence $\ell_{\gamma}(X)$ only depends on the free homotopy class of $\gamma$).

If $\gamma$ is homotopic to a boundary $\beta$ of $X$ (or a cusp), then $\ell_{\gamma}(X)$ is defined as the length of $\beta$ (and $\ell_{\gamma}(X) = 0$ if $\gamma$ is homotopic to a cusp).

More generally, every family $(\gamma_1, \dots, \gamma_k)$ of closed essential curves on a hyperbolic surface $X$ is freely homotopic to a unique family of geodesics. If the curves $\gamma_1, \dots, \gamma_k$ are simple, disjoint, and if for any $1 \le i \le k$, $\gamma_i$ is not homotopic to any of the $\gamma_j$'s or $\gamma_j^{-1}$'s for $j \ne i$, we say that $(\gamma_1, \dots, \gamma_k)$ is a multi-curve, and its geodesic representative is a multi-geodesic. If $\Gamma = (\gamma_1, \dots, \gamma_k)$ is a family of curves on $X$, we define the length of $\Gamma$ to be the vector:

$$ \ell_{\Gamma}(X) = (\ell_{\gamma_1}(X), \dots, \ell_{\gamma_k}(X)), \qquad (3) $$

i.e. $\ell_{\Gamma}(X)$ is the length of the geodesic representative of $\Gamma$. Notation (3) also makes sense if some of the components of $\Gamma$ are homotopic to a boundary or a cusp of $X$.

If $\Gamma$ is a family of curves on $S_{g,n}$ and $(X, f)$ is a marked surface, we define:

$$ \ell_{\Gamma}(X, f) = \ell_{f(\Gamma)}(X). $$

This expression induces a well-defined function on the Teichmüller space $\mathcal{T}_{g,n}(L)$.

The mapping class group acts on the free homotopy classes of closed curves (or families of curves) on $S_{g,n}$, and we can therefore consider the orbits of this action. If $\Gamma$ is a family of closed curves on $S_{g,n}$, we denote by $\mathcal{O}_{\Gamma}$ the orbit of its free homotopy class.

Let $F : \mathbb{R}^k \to \mathbb{C}$ be a test function (e.g. compactly supported). The set of lengths of closed geodesics of a given hyperbolic surface is a discrete subset of $\mathbb{R}_{>0}$, thus we can define:

$$ F^{\Gamma}: X \in \mathcal{M}_{g,n}(L) \mapsto \sum_{[\Gamma'] \in \mathcal{O}_{\Gamma}} F \circ \ell_{\Gamma'}(X). \qquad (4) $$

Each of the terms of the sum taken individually is ill-defined on $\mathcal{M}_{g,n}(L)$ but the sum is well-defined. Indeed, $\ell_{\Gamma'}(X, f)$ makes sense only for an element $(X, f)$ of the Teichmüller space, not for an element $X$ of $\mathcal{M}_{g,n}(L)$, since the mapping class group can send the homotopy class of $\Gamma'$ to multiple distinct free homotopy classes of curves on $S_{g,n}$. However considering the sum over the whole orbit makes it invariant under the action of the mapping class group. Thus the sum is well-defined on $\mathcal{M}_{g,n}(L)$. We can therefore integrate it on $\mathcal{M}_{g,n}(L)$.

Random variables of the form $F^{\Gamma}$ provide a natural tool to count objects on hyperbolic surfaces. This is the reason why we focus on integrating them on the moduli space. The following example illustrates this fact.
*Example.* Given a closed curve $\gamma$ on $S_{g,n}$ and $a > 0$, we retrieve the random variable $N_{\gamma}(a)$ by taking $F = \mathbb{1}_{[0,a]}$ in (4).

Mirzakhani developed in [Mir07b] an integration formula for random variables of the form $F^{\Gamma}$, where $\Gamma$ is a multi-curve.

**Theorem 1.4** (Mirzakhani's integration formula, [Mir07b]). Let $\Gamma = (\gamma_1, \dots, \gamma_k)$ be a multi-curve on $S_{g,n}$. For every test function $F: \mathbf{R}^k \to \mathbf{C}$, we have:

$$ \mathbb{E}[F^{\Gamma}] = \int_{\mathbf{R}_{\ge 0}^k} F(x_1, \dots, x_k) x_1 \cdots x_k V_{\Gamma}(x_1, \dots, x_k, L) \frac{dx_1 \cdots dx_k}{V_{g,n}(L)}, $$

where $V_{\Gamma}(x_1, \dots, x_k, L)$ is the volume of the moduli space of the complement $S_{g,n} \setminus \Gamma$ of the multi-curve $\Gamma$, where the lengths of the two boundary components of $S_{g,n} \setminus \Gamma$ corresponding to $\gamma_i$ are given by $x_i$, for all $1 \le i \le k$. If $S_{g,n} \setminus \Gamma$ is disconnected with connected components $S_1, \dots, S_m$, the volume $V_{\Gamma}(x_1, \dots, x_k, L)$ is the product of the volumes of the moduli spaces of $S_1, \dots, S_m$.

As a direct consequence of this integration formula and Theorem 1.3 we have the following conclusion:

**Corollary 1.5.** If $\gamma$ is a simple closed curve, then $\mathbb{E}[N_{\gamma}(a)]$ is a polynomial of degree $6g - 6 + 2n$ in the variable $a$.

Sapir obtained bounds on the number of geodesics of bounded length and bounded number of self-intersections on any closed hyperbolic surface in [Sap16a] and [Sap16b]. Mirzakhani also provided an asymptotic equivalent of $N_{\gamma}(a)$ when $\gamma$ is not simple.

**Theorem 1.6** ([Mir16]). Let $\gamma$ be a closed curve on $S_{g,n}$. There exist a constant $n_{\gamma} \in \mathbb{Q}$ and a smooth and integrable function $B: \mathcal{M}_{g,n}(L) \to \mathbf{R}_{\ge 0}$ such that:

$$ N_{\gamma}(a)(X) \underset{a \to \infty}{\sim} n_{\gamma} \frac{B(X)}{b_{g,n}} a^{6g-6+2n}, $$

for all $X \in \mathcal{M}_{g,n}(L)$, where $b_{g,n} = \int_{\mathcal{M}_{g,n}(L)} B(X) d\mu_{WP}(X)$.

*Remark.* The constant $n_{\gamma}$ depends on the topology of the surface $S_{g,n}$, i.e. on $(g, n)$.

Mirzakhani also managed to prove in [Mir16] that dominated convergence can be used in order to integrate the equivalent provided by Theorem 1.6 and obtain the following result:

**Corollary 1.7** ([Mir16]). With the same notations as in Theorem 1.6, we have:

$$ \mathbb{E}[N_{\gamma}(a)] \underset{a \to \infty}{\sim} \frac{n_{\gamma}}{V_{g,n}(L)} a^{6g-6+2n}. $$

Erlandsson and Souto proved a similar, more precise result for geodesics on once-holed tori in [ES16]. Parlier, Erlandsson and Souto in [EPS20], Erlandsson and Souto in [ES24] and Trin in [Tri24] proved results going in the same direction for other notions of the length of a curve.

The aim of this article is to develop an integration formula for eight-shaped curves (Theorem 1.1) in order to obtain a more detailed asymptotic expansion of the average counting function $\mathbb{E}[N_{\gamma}(a)]$ when $a \to \infty$, when $\gamma$ has one self-intersection. Our Corollary 1.2 improves the asymptotic equivalent provided by Corollary 1.7 for such curves. However our method is different from the one used in [Mir16] and we do not know if it is possible to improve the pointwise result given in Theorem 1.6 by using our method. Indeed, we use the tools developed by Anantharaman and Monk in [AM25] instead of averaging a pointwise result on $\mathcal{M}_{g,n}(L)$. The method we use in this article is only suited for eight-shaped curves, and even some orbits of such curves provide some illustration of the limits of our method. Our further goal will be to generalize Theorem 1.1 and Corollary 1.2 for curves filling a pair of pants in a first instance,
and for even more general curves in following steps, but in order to do so we shall modify some of the arguments we use in this article.

We also discuss the continuity of $\ell \mapsto V_O(\ell, L)$ and give an explicit expression of this function for some specific MCG$_{g,n}$-orbits of homotopy classes of eight-shaped curves.

### 1.4. Motivations and links with the work of Anantharaman and Monk.
The Selberg trace formula [Sel56] establishes a link between the lengths of closed geodesics on a hyperbolic surface and the eigenvalues of the Laplace operator on this surface, hence having a better understanding of the counting functions $N_\gamma(a)$ provides a better understanding of the spectral geometry of hyperbolic surfaces. Indeed, denoting by $\lambda_1 > 0$ the smallest non-zero eigenvalue of the Laplace operator, Anantharaman and Monk proved in [AM24] by studying curves filling a surface of Euler characteristic -1, that when $g \to \infty$, for any $\varepsilon > 0$ the Weil-Petersson probability of having $\lambda_1 \ge \frac{2}{9} - \varepsilon$ tends to 1 on closed hyperbolic surfaces of genus $g$. They improved this result in [AM25] by proving that the probability of having $\lambda_1 \ge \frac{1}{4} - \varepsilon$ tends to 1. Studying functions of the form $V_O(\cdot, L)$ may similarly help us understanding the behavior of the eigenvalues of the Laplace operator at fixed genus and number of boundaries.

Since the MCG$_{g,n}$-orbit $O_\gamma$ of a closed curve $\gamma$ depends on the signature $(g, n)$ of the base surface $S_{g,n}$, considering functions of the form $F^\gamma$ does not really make sense when taking $g \to \infty$. To remedy this problem, Anantharaman and Monk defined in [AM24] the notion of local topological type of a curve on a closed surface. Given a local topological type $T$ (e.g. the type “simple” or the type “eight-shaped”), Anantharaman and Monk considered the $T$-average of a test function $F: \mathbf{R} \to \mathbf{R}$ on $M_g$ (where $M_g$ is the moduli space of closed surfaces of genus $g$):

$$ \langle F \rangle_g^T = \mathbb{E} \left[ \sum_{\substack{\gamma \in \mathcal{G}(X) \\ \gamma \sim T}} F \circ \ell_\gamma \right]. $$

Here $\mathcal{G}(X)$ is the set of closed geodesics on $X \in M_g$ and $\gamma \sim T$ means that $\gamma$ has local topological type $T$, and the set $\{\gamma \in \mathcal{G}(X) | \gamma \sim T\}$ is partitionned into a finite number (growing with $g$) of mapping class group orbits of geodesics on $X$. Anantharaman and Monk proved in [AM24] that for any local topological type $T$, there exists a unique locally integrable function $V_g^T: \mathbf{R}_{\ge 0} \to \mathbf{R}_{\ge 0}$ such that for every test function $F: \mathbf{R} \to \mathbf{R}$, we have:

$$ \langle F \rangle_g^T = \int_0^\infty F(\ell) V_g^T(\ell) \frac{d\ell}{V_g}. $$

$V_g^T$ is called the volume function associated with local type $T$ on surfaces of genus $g$. By using the same argument we can deduce the existence of the function $V_O(\cdot, L)$ in our Theorem 1.1. Anantharaman and Monk studied in [AM24; AM25] the large genus behavior of the functions $V_g^T$ but they did not obtain any information about these volume functions for fixed $g$. On the opposite, we study the asymptotical behavior of $V_O(\ell, L)$ when $\ell \to \infty$ for fixed orbits $O$ of curves, on surfaces with fixed genus and boundaries.

Due to the fact that in this article we work with surfaces of fixed topology, it seems more satisfactory to fix MCG$_{g,n}$-orbits and thus we will be more interested in functions of the form $\mathbb{E}[F^\gamma]$ rather than $\langle F \rangle_g^T$.

### 1.5. Pseudo-convolution.
The main tool which will be used in this paper, aside from Mirzakhani's integration formula, is the pseudo-convolution. This operation on locally integrable functions was developed by Anantharaman and Monk in [AM25] in order to compute the asymptotical behavior at high genus of the volume functions $V_g^T$, in order to estimate the spectral gap of the Laplace-Beltrami operator on hyperbolic surfaces.

We will recall from [AM25] the definition of the pseudo-convolution and adapt the results of Anantharaman and Monk in order for us to prove our main result.
We shall also recall from [AM24] the definition of *Friedman-Ramanujan functions*, i.e. functions of the form:

$$f(l) = e^l p(l) + O((1+l)^c e^{l/2}),$$

where $p$ is a polynomial and $c > 0$ is a constant (see Definition 3.1). Here the $O$ can (and will) be taken in a weak sense which will be explained later. Estimate (2) can therefore be re-expressed by saying that $l \mapsto e^l V_{O_\gamma}(l, L)$ is a *Friedman-Ramanujan function in the weak sense*.

**1.6. Acknoledgments.** This research has been partially funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 101096550).

I would also like to thank Nalini Anantharaman and Arnaud Maret for their helpful comments during the preparation of this article.

## 2. AN INTEGRATION FORMULA FOR EIGHT-SHAPED GEODESICS

**2.1. The surface filled by an eight-shaped curve.** Let us recall the definition of the surface filled by a curve.

**Definition 2.1.** Let $\gamma$ be a closed curve in minimal position on a hyperbolic surface $X$. The surface filled by $\gamma$, denoted by $S(\gamma)$, is defined using the following process:

(i) we consider $\varepsilon > 0$ small enough so the regular neighborhood $V_\varepsilon(\gamma) = \{x \in X | d(x, \gamma) < \varepsilon\}$ retracts to $\gamma$;

(ii) denoting $C_1, \dots, C_k$ the connected components of $X \setminus V_\varepsilon$ homeomorphic to a disc, we define:

$$S(\gamma) = V_{\varepsilon} \cup \bigcup_{i=1}^{k} C_{i}.$$

*Remark.* If we want to consider the surface filled by a closed curve on the base surface $S_{g,n}$, we need to endow $S_{g,n}$ with a hyperbolic metric (so that the regular neighborhood $V_\varepsilon(\gamma)$ is well-defined). However the surface filled by a closed curve is defined up to isotopy, and it does not depend on the choice of this metric.

**Lemma 2.2.** Let $\gamma$ be an eight-shaped curve in minimal position. The surface filled by $\gamma$ is a pair of pants.

From now on, all the curves we consider will be in minimal position.

Let $\gamma$ be an eight-shaped curve on $S_{g,n}$, self-intersecting at the point $p$. Let us denote by $\alpha, \alpha'$ and $\beta$ the connected components of $\partial S(\gamma)$ such that the 2-curve $(\alpha, \alpha')$ is freely homotopic to the separating resolution of $\gamma$ at $p$ and $\beta$ is homotopic to the non-separating resolution of $\gamma$ at $p$ (see Figure 1). We orient the curves $\alpha, \alpha'$ and $\beta$ so that $S(\gamma)$ is on the left of these curves.

<figure><img src="image_2.png" /><figcaption>FIGURE 1. The surface filled by $\gamma$.</figcaption></figure>

There are two possible scenarios:
* the free homotopy classes $[\alpha]$, $[\alpha']$ and $[\beta]$ are distinct, in which case we shall say that $\gamma$ exactly fills a pair of pants;

* two of these free homotopy classes are equal, in which case we shall say that $\gamma$ almost fills a holed torus.

**Lemma 2.3** ([RY14]). Let $\gamma$ be an eight-shaped geodesic on a hyperbolic surface $X$ and let $(\alpha, \alpha')$ be the separating resolution of $\gamma$ at its self-intersecting point, and $\beta$ its non-separating resolution (i.e. $\alpha, \alpha'$ and $\beta$ are defined as on Figure 1). We have:

$$ \cosh\left(\frac{\ell_{\gamma}(X)}{2}\right) = 2\cosh\left(\frac{\ell_{\alpha}(X)}{2}\right)\cosh\left(\frac{\ell_{\alpha'}(X)}{2}\right) + \cosh\left(\frac{\ell_{\beta}(X)}{2}\right). $$

*Remark.* Since at least one of the geodesics $\alpha, \alpha'$ and $\beta$ is not a cusp of $X$, we have $\ell_{\gamma}(X) > 2\arg\cosh(3)$.

Hence we define:

$$ h : (x, y, z) \in \mathbb{R}_{>0}^3 \mapsto 2\arg\cosh\left(2\cosh\left(\frac{x}{2}\right)\cosh\left(\frac{y}{2}\right) + \cosh\left(\frac{z}{2}\right)\right). $$

**2.2. Description of the orbits of an eight-shaped curve.** By using the separating and non-separating resolutions of an eight-shaped curve, we are able to describe the topological type of such a curve in terms of orbits of a 3-curve.

**Lemma 2.4.** Let $\gamma$ be an eight-shaped curve on $S_{g,n}$, and let $\alpha, \alpha'$ and $\beta$ be defined as in Figure 1. Let $\Gamma = (\alpha, \alpha', \beta)$. We have $F^{\gamma} = (F \circ h)^{\Gamma}$.

*Proof.* On the one hand let $\Gamma' = (\alpha_1, \alpha_2, \alpha_3)$ be a 3-curve on $S_{g,n}$ such that $\Gamma'$ can be identified with the boundary $\partial S$ of a pair of pants $S \subset S_{g,n}$, oriented such that $S$ is on the left of each component of $\partial S$. Let us denote by $\Psi(\Gamma')$ the free homotopy class of the eight-shaped curve which loops around $\alpha_1$ and $\alpha_2$, such that one of its loop is homotopic to $\alpha_1$ and the other is homotopic to $\alpha_2^{-1}$.

The homotopy class $\Psi(\Gamma')$ only depends on $[\Gamma']$, thus $\Psi$ induces a well-defined map $\bar{\Psi}$ between the set of free homotopy classes of such 3-curves and the set of free homotopy classes of eight-shaped curves.

On the other hand if $\gamma'$ is an eight-shaped curve on $S_{g,n}$, let us denote by $\beta_1, \beta_2, \beta_3$ the connected components of $\partial S(\gamma')$, oriented such that $S(\gamma')$ is on the left of these curves, and labelled such that one of the loops of $\gamma'$ is homotopic to $\beta_1$, and the other loop of $\gamma'$ is homotopic to $\beta_2^{-1}$. Let $\Phi(\gamma') = [\beta_1, \beta_2, \beta_3]$. Same as before, $\Phi(\gamma')$ only depends on $[\gamma']$ and $\Phi$ induces a well-defined map $\bar{\Phi}$ on the set of free homotopy class of eight-shaped curves.

By construction $\bar{\Psi}$ and $\bar{\Phi}$ are inverse, hence when restricted to an MCG$_{g,n}$-orbit, they provide bijections between $\mathcal{O}_{\gamma}$ and $\mathcal{O}_{\Gamma}$. By using Lemma 2.3, we therefore have for any test function $F$ and any marked hyperbolic surface $(X, f)$:

$$ \sum_{[\gamma'] \in \mathcal{O}_{\gamma}} F(\ell_{\gamma'}(X, f)) = \sum_{[\Gamma'] \in \mathcal{O}_{\Gamma}} F \circ h(\ell_{\Gamma'}(X, f)). $$

Hence $F^{\gamma} = (F \circ h)^{\Gamma}$ on $M_{g,n}(L)$. $\square$

*Remark.* Anantharaman and Monk provided in [AM24] an example of curve filling a 4-holed sphere for which the map $\bar{\Psi}$ is well-defined, i.e. given a 4-curve $\Gamma$ which bounds a 4-holed sphere $S \subset S_{g,n}$, $\bar{\Psi}([\Gamma])$ is the homotopy class of some fixed curve on $S$. However for this example, the map $\bar{\Psi}$ is not a bijection anymore, but it is still an equivariant map, thus $\#\bar{\Psi}^{-1}([\gamma])$ only depends on $\mathcal{O}_{\gamma}$. Denoting $\bar{\Psi}([\Gamma]) = [\gamma]$ and $m(\gamma) = \#\bar{\Psi}^{-1}([\gamma])$, we therefore have for all marked hyperbolic surface $(X, f)$:

$$ \sum_{[\Gamma'] \in \mathcal{O}_{\Gamma}} F(\ell_{\bar{\Psi}([\Gamma'])}(X, f)) = m(\gamma) \sum_{[\gamma'] \in \mathcal{O}_{\gamma}} F(\ell_{\gamma}(X, f)). $$

We can divide the orbits of eight-shaped curves into 9 types. Let $\gamma$ be such a curve, we say that $\gamma$ is of type:
* ($\mathcal{P}_0$) if $\gamma$ exactly fills a pair of pants and if the curves $\alpha, \alpha'$ and $\beta$ are essential;

* ($\mathcal{P}_1$) if $\gamma$ exactly fills a pair of pants and if exactly one of the curves $\alpha$ and $\alpha'$ is essential and $\beta$ is essential;

* ($\mathcal{P}'_1$) if $\gamma$ exactly fills a pair of pants and if $\beta$ is non-essential and $\alpha, \alpha'$ are essential;

* ($\mathcal{P}_2$) if $\gamma$ exactly fills a pair of pants and if $\alpha, \alpha'$ are not essential and $\beta$ is essential;

* ($\mathcal{P}'_2$) if $\gamma$ exactly fills a pair of pants and if exactly one of the curves $\alpha$ and $\alpha'$ is essential and $\beta$ is not essential;

* ($\mathcal{T}_0$) if $\gamma$ almost fills a holed torus and if $[\alpha] = [\alpha']$ and $\beta$ is essential;

* ($\mathcal{T}'_0$) if $\gamma$ almost fills a holed torus and if $[\alpha] = [\beta]$ and $\alpha'$ is essential or if $[\alpha'] = [\beta]$ and $\alpha$ is essential;

* ($\mathcal{T}_1$) if $\gamma$ almost fills a holed torus and if $[\alpha] = [\alpha']$ and $\beta$ is not essential;

* ($\mathcal{T}'_1$) if $\gamma$ almost fills a holed torus and if $[\alpha] = [\beta]$ and $\alpha'$ is not essential or if $[\alpha'] = [\beta]$ and $\alpha$ is not essential;

Remark. If $\gamma$ is of type ($\mathcal{T}_1$) or ($\mathcal{T}'_1$), then $g = n = 1$, i.e. $S_{g,n}$ is a holed torus (also called a handle).

2.3. **Integration formula.** By using Corollary 2.4 and Mirzakhani’s integration formula, we can obtain an integration formula for eight-shaped curves.

**Corollary 2.5.** Let $\gamma$ be an eight-shaped curve on $S_{g,n}$. There exists a polynomial $Q_{\gamma}^{g,n,L}$ of at most three variables, depending only on the orbit of $\gamma$, such that for every test function $F: \mathbf{R} \to \mathbf{C}$:

(i) if $\gamma$ is of type ($\mathcal{P}_0$), then $Q_{\gamma}^{g,n,L}$ has three variables and:

$$ \mathbb{E}[F^{\gamma}] = \int_{\mathbf{R}_{>0}^3} F(h(x, y, z)) Q_{\gamma}^{g,n,L}(x, y, z) \frac{dx dy dz}{V_{g,n}(L)}; $$

(ii) if $\gamma$ is of type ($\mathcal{P}_1$), then $Q_{\gamma}^{g,n,L}$ has two variables and there exists $1 \le i \le n$ such that:

$$ \mathbb{E}[F^{\gamma}] = \int_{\mathbf{R}_{>0}^2} F(h(x, L_i, z)) Q_{\gamma}^{g,n,L}(x, z) \frac{dx dz}{V_{g,n}(L)}; $$

(iii) if $\gamma$ is of type ($\mathcal{P}'_1$), then $Q_{\gamma}^{g,n,L}$ has two variables and there exists $1 \le i \le n$ such that:

$$ \mathbb{E}[F^{\gamma}] = \int_{\mathbf{R}_{>0}^2} F(h(x, y, L_i)) Q_{\gamma}^{g,n,L}(x, y) \frac{dx dy}{V_{g,n}(L)}; $$

(iv) if $\gamma$ is of type ($\mathcal{P}_2$), then $Q_{\gamma}^{g,n,L}$ has one variable and there exist $1 \le i, j \le n, i \ne j$, such that:

$$ \mathbb{E}[F^{\gamma}] = \int_{\mathbf{R}_{>0}^2} F(h(L_i, L_j, z)) Q_{\gamma}^{g,n,L}(z) \frac{dz}{V_{g,n}(L)}; $$

(v) if $\gamma$ is of type ($\mathcal{P}'_2$), then $Q_{\gamma}^{g,n,L}$ has one variable and there exist $1 \le i, j \le n, i \ne j$, such that:

$$ \mathbb{E}[F^{\gamma}] = \int_{\mathbf{R}_{>0}^2} F(h(x, L_i, L_j)) Q_{\gamma}^{g,n,L}(x) \frac{dx}{V_{g,n}(L)}; $$

(vi) if $\gamma$ is of type ($\mathcal{T}_0$), then $Q_{\gamma}^{g,n,L}$ has two variables and:

$$ \mathbb{E}[F^{\gamma}] = \int_{\mathbf{R}_{>0}^2} F(h(x, x, z)) Q_{\gamma}^{g,n,L}(x, z) \frac{dx dz}{V_{g,n}(L)}; $$

(vii) if $\gamma$ is of type ($\mathcal{T}'_0$), then $Q_{\gamma}^{g,n,L}$ has two variables and:

$$ \mathbb{E}[F^{\gamma}] = \int_{\mathbf{R}_{>0}^2} F(h(x, y, x)) Q_{\gamma}^{g,n,L}(x, y) \frac{dx dy}{V_{g,n}(L)}; $$
(viii) if $\gamma$ is of type $(\mathcal{T}_1)$, then $Q_{\gamma}^{1,1,L_1}(x) = x$ and:

$$ \mathbb{E}[F^{\gamma}] = \int_{\mathbf{R}_{>0}} F(h(x, x, L_1)) Q_{\gamma}^{1,1,L_1}(x) \frac{dx}{V_{1,1}(L_1)}; $$

(ix) if $\gamma$ is of type $(\mathcal{T}_1')$, then $Q_{\gamma}^{1,1,L_1}(x) = x$ and:

$$ \mathbb{E}[F^{\gamma}] = \int_{\mathbf{R}_{>0}} F(h(L_1, x, x)) Q_{\gamma}^{1,1,L_1}(x) \frac{dx}{V_{1,1}(L_1)}; $$

Moreover if $Q_{\gamma}^{g,n,L}$ is a polynomial of $k \in \{1, 2, 3\}$ variables, then we have:

$$ Q_{\gamma}^{g,n,L}(x_1, \dots, x_k) = x_1 \cdots x_k V_{\Gamma_{\gamma}}(x_1, \dots, x_k, L) \in \mathbf{R}_{6g-6+2n-k} [x_1, \dots, x_k], $$

where $V_{\Gamma_{\gamma}}(x_1, \dots, x_k, L)$ is the volume of the moduli space of the surface obtained from $S_{g,n}$ by cutting along $\Gamma_{\gamma}$ with $\ell_{\Gamma_{\gamma}} = (x_1, \dots, x_k)$.

This integration result makes it possible for us to compute the coefficient $n_{\gamma}$ appearing in Theorem 1.6 for eighth-shaped curves.

**Corollary 2.6.** With the same notations as above, let us denote by:

$$ V_{\Gamma}(x_1, \dots, x_k, L) = \sum_{i_1+\dots+i_k \le 3g-3+n-k} C_{i_1 \dots i_k}^{\Gamma}(L) x_1^{2i_1} \dots x_k^{2i_k}. $$

The leading coefficients of $(x_1, \dots, x_k) \mapsto V_{\Gamma}(x_1, \dots, x_k, L)$, i.e. the coefficients $C_{i_1 \dots i_k}^{\Gamma}$ with $i_1 + \dots + i_k = 3g - 3 + n - k$, do not depend on $L$, and if $(g, n) \ne (1, 1)$, then $n_{\gamma}$ is the leading term of the polynomial function:

$$ a \mapsto \sum_{i_1+\dots+i_k=3g-3+n-k} C_{i_1 \dots i_k}^{\Gamma} I_{i_1 \dots i_k}(a), $$

where:

(i) if $\gamma$ is of type $(\mathcal{P}_0)$, then $k=3$ and:

$$ I_{i_1 i_2 i_3}(a) = \int_{\mathbb{R}_{>0}^3} \mathbb{1}_{\{\max\{x+y,z\} \le a\}} x^{2i_1+1} y^{2i_2+1} z^{2i_3+1} dx dy dz = \frac{(2i_1+1)!(2i_2+1)!}{2(i_3+1)(2(i_1+i_2+2))!} a^{2(i_1+i_2+i_3+3)}; $$

(ii) if $\gamma$ is of type $(\mathcal{P}_1)$, then $k=2$ and there exists $1 \le i \le n$ such that:

$$ I_{i_1 i_2}(a) = \int_{\mathbb{R}_{>0}^2} \mathbb{1}_{\{\max\{x+L_i, z\} \le a\}} x^{2i_1+1} z^{2i_2+1} dx dz \sim_{a \to \infty} \frac{a^{2(i_1+i_2+2)}}{4(i_1+1)(i_2+1)}; $$

(iii) if $\gamma$ is of type $(\mathcal{P}_1')$, then $k=2$ and there exists $1 \le i \le n$ such that:

$$ I_{i_1 i_2}(a) = \int_{\mathbb{R}_{>0}^2} \mathbb{1}_{\{\max\{x+y, L_i\} \le a\}} x^{2i_1+1} y^{2i_2+1} dx dy = \frac{(2i_1+1)!(2i_2+1)!}{(2(i_1+i_2+2))!} a^{2(i_1+i_2+2)}; $$

(iv) if $\gamma$ is of type $(\mathcal{P}_2)$, then $k=1$ and there exist $1 \le i, j \le n$ such that:

$$ I_{i_1}(a) = \int_{\mathbb{R}_{>0}^2} \mathbb{1}_{\{\max\{L_i+L_j, z\} \le a\}} z^{2i_1+1} dz = \frac{a^{2(i_1+1)}}{2(i_1+1)}; $$

(v) if $\gamma$ is of type $(\mathcal{P}_2')$, then $k=1$ and there exist $1 \le i, j \le n$ such that:

$$ I_{i_1}(a) = \int_{\mathbb{R}_{>0}^2} \mathbb{1}_{\{\max\{x+L_i, L_j\} \le a\}} x^{2i_1+1} dx \sim_{a \to \infty} \frac{a^{2(i_1+1)}}{2(i_1+1)}; $$

(vi) if $\gamma$ is of type $(\mathcal{T}_0)$, then $k=2$ and:

$$ I_{i_1 i_2}(a) = \int_{\mathbb{R}_{>0}^2} \mathbb{1}_{\{\max\{2x,z\} \le a\}} x^{2i_1+1} z^{2i_2+1} dx dz = \frac{a^{2(i_1+i_2+2)}}{4^{i_1+3}(i_1+1)(i_2+1)}; $$
(vii) if $\gamma$ is of type $(\mathcal{T}'_0)$, then $k=2$ and:

$$I_{i_1 i_2}(a) = \int_{\mathbf{R}_{\ge 0}^2} 1_{\{x+y \le a\}} x^{2i_1+1} y^{2i_2+1} dx dy = \frac{(2i_1+1)!(2i_2+1)!}{(2(i_1+i_2+2))!} a^{2(i_1+i_2+2)}.$$

Moreover if $(g, n) = (1, 1)$:

(viii) if $\gamma$ is of type $(\mathcal{T}_1)$, we have $n_\gamma = \frac{1}{8}$;

(ix) if $\gamma$ is of type $(\mathcal{T}'_1)$, we have $n_\gamma = \frac{1}{2}$.

**Proof.** By using Theorem 1.3 we can deduce the fact that the leading coefficients $C_{i_1 \dots i_k}^{\Gamma}$ do not depend on $L$.

Moreover, according to Lemma 2.3, there exist constants $c_1, c_2 > 0$ such that for all $(x, y, z) \in \mathbf{R}_{\ge 0}^3$:

$$\max\{x + y, z\} - c_1 \le h(x, y, z) \le \max\{x + y, z\} + c_2.$$

Let us assume for instance that $\gamma$ is of type $(\mathcal{P}_0)$. In this case we have:

$$Q_{\gamma}^{g,n,L}(x, y, z) = \sum_{i_1+i_2+i_3 \le 3g-6+n} C_{i_1 i_2 i_3}^{\Gamma}(L) x^{2i_1+1} y^{2i_2+1} z^{2i_3+1},$$

hence if $a \ge 0$:

$$\sum_{i_1+i_2+i_3 \le 3g-6+n} \frac{C_{i_1 i_2 i_3}^{\Gamma}(L)}{V_{g,n}(L)} I_{i_1 i_2 i_3}(a-c_2) \le \mathbb{E}[N_{\gamma}(a)] \le \sum_{i_1+i_2+i_3 \le 3g-6+n} \frac{C_{i_1 i_2 i_3}^{\Gamma}(L)}{V_{g,n}(L)} I_{i_1 i_2 i_3}(a+c_1).$$

Since $I_{i_1 i_2 i_3}$ is a polynomial of degree $2i_1 + 2i_2 + 2i_3 + 6$, we have:

$$\mathbb{E}[N_{\gamma}(a)] \underset{a \to \infty}{\sim} \sum_{i_1+i_2+i_3=3g-6+n} \frac{C_{i_1 i_2 i_3}^{\Gamma}(L)}{V_{g,n}(L)} I_{i_1 i_2 i_3}(a).$$

We can therefore conclude by using Theorem 1.6 that:

$$n_{\gamma} a^{6g-6+2n} \underset{a \to \infty}{\sim} \sum_{i_1+i_2+i_3=3g-6+n} C_{i_1 i_2 i_3}^{\Gamma} I_{i_1 i_2 i_3}(a).$$

By repeating the same process with every type of eight-shaped curve, we can conclude. $\square$

**Example.** If $(g, n) = (0, 4)$, i.e. if $S_{g,n}$ is a four-holed sphere, then $\gamma$ is either of type $(\mathcal{P}_2)$ or $(\mathcal{P}'_2)$ and $n_\gamma = \frac{1}{2}$. More generally, by using Mirzakhani's results on the leading coefficients of $V_{0,n}$ from [Mir07b], if $\gamma$ is a curve of type $(\mathcal{P}_2)$ or $(\mathcal{P}'_2)$ on $S_{0,n}$, we have:

$$n_{\gamma} = \frac{1}{2^{n-3}(n-3)!}.$$

If $\gamma$ is of type $(\mathcal{P}_1)$ (respectively $(\mathcal{P}'_1)$) on $S_{0,n}$, then we have $n \ge 5$ and there exist two integers $n_1, n_2 \ge 2$ such that $n_1 + n_2 = n - 1$ and:

$$n_{\gamma} = \frac{1}{2^{n-3}(n_1-1)!(n_2-1)!}$$

(respectively:

$$n_{\gamma} = \frac{(2(n_1-1))!(2(n_2-1))!}{2^{n-3}(n_1-1)!(n_2-1)!(2(n-3))!}.$$

Finally, if $\gamma$ is of type $(\mathcal{P}_0)$ on $S_{g,n}$, then we have $n \ge 6$ and there exist integers $n_1, n_2, n_3 \ge 2$ such that $n_1 + n_2 + n_3 = n$ and:

$$n_{\gamma} = \frac{(2(n_1-1))!(2(n_2-1))!}{2^{n-3}(n_1-1)!(n_2-1)!(n_3-1)!(2(n_1+n_2-2))!}.$$
2.4. **Link with convolution.** Let us consider for example a curve $\gamma$ of type ($\mathcal{P}_0$). Our goal is to write:

$$ \mathbb{E}[F^{\gamma}] = \int_{\mathbf{R}_{>0}^3} F(h(x, y, z)) Q_{\gamma}^{g,n,L}(x, y, z) dxdydz = \int_{0}^{\infty} F(\ell) V_{\mathcal{O}_{\gamma}}(\ell, L) \frac{d\ell}{V_{g,n}(L)}, \quad (5) $$

and to study the asymptotic behavior of $V_{\mathcal{O}_{\gamma}}(\ell, L)$ as $\ell \to \infty$. If we had been able to separate the variables in order to write $Q_{\gamma}^{g,n,L}(x, y, z) = f_1(x)f_2(y)f_3(z)$ and if $h(x, y, z) = x + y + z$, then we would have:

$$ \mathbb{E}[F^{\gamma}] = \int_{\mathbf{R}_{>0}^3} F(x + y + z) f_1(x) f_2(y) f_3(z) \frac{dxdydz}{V_{g,n}(L)}. $$

Disintegrating along the level sets of $(x, y, z) \mapsto x + y + z$, we would have:

$$ \mathbb{E}[F^{\gamma}] = \int_{0}^{\infty} F(\ell) \left( \int_{\ell = x+y+z} f_1(x) f_2(y) f_3(z) \mathbf{1}_{\mathbf{R}_{>0}^3} (x, y, z) \frac{dxdydz}{d\ell} \right) \frac{d\ell}{V_{g,n}(L)}, $$

where $\frac{dxdydz}{d\ell}$ is the disintegration of the Lebesgue measure on the plane $\{\ell = x + y + z\}$. Thus, under these very specific assumptions on $Q_{\gamma}^{g,n,L}$ and $h$, we would have:

$$ \mathbb{E}[F^{\gamma}] = \int_{0}^{\infty} F(\ell) f_1 * f_2 * f_3(\ell) \bigg|_{\mathbf{R}_{>0}^3} \frac{d\ell}{V_{g,n}(L)}, $$

where $f_1 * f_2 * f_3|_{\mathbf{R}_{>0}^3}$ is the usual convolution on $\mathbf{R}_{>0}^3$, i.e. $V_{\mathcal{O}_{\gamma}}$ would have been defined by the usual convolution of $f_1, f_2$ and $f_3$.

Unfortunately, $Q_{\gamma}^{g,n,L}$ and $h$ do not satisfy these assumptions. However $Q_{\gamma}^{g,n,L}$ is a polynomial, hence by linearity we can assume that it separates the variables, and we will see in what follows that far from some planes in $\mathbf{R}_{>0}^3$, the function $h$ behaves similarly to linear forms. Hence we can see $V_{\mathcal{O}_{\gamma}}$ as a convolution product which is somehow modified by the input of $h$. We will develop this in Paragraph 3.2.

### 3. PSEUDO-CONVOLUTION OF FRIEDMAN-RAMANUJAN FUNCTIONS

**3.1. Friedman-Ramanujan functions.** Let us recall first the definition of Friedman-Ramanujan functions from [AM24].

**Definition 3.1 ([AM24]).** A locally integrable function $f: \mathbf{R}_{>0} \to \mathbf{C}$ belongs to the class $\mathcal{R}$ if there exist constants $c, c_1 > 0$ such that:

$$ \forall \ell > 0, |f(\ell)| \le c_1 (\ell + 1)^c e^{\frac{\ell}{2}}. $$

The space $\mathcal{R}_w$ is defined to be the space of measurable functions $f: \mathbf{R}_{>0} \to \mathbf{C}$ such that there exist constants $c, c_1$ verifying:

$$ \forall n \ge 1, \int_{0}^{n} |f(\ell)| d\ell \le c_1 (n + 1)^c e^{\frac{n}{2}}. $$

The space of Friedman-Ramanujan functions is the space $\mathcal{F} = e^{\ell}\mathbf{C}[\ell] \oplus \mathcal{R}$. The degree of a Friedman-Ramanujan function is the degree of its polynomial part.

The space of weak Friedman-Ramanujan functions is the space $\mathcal{F}_w = e^{\ell}\mathbf{C}[\ell] \oplus \mathcal{R}_w$. If $m \ge 0$, we denote $\mathcal{F}^m = e^{\ell}\mathbf{C}_{m-1}[\ell] \oplus \mathcal{R}$ and $\mathcal{F}_w^m = e^{\ell}\mathbf{C}_{m-1}[\ell] \oplus \mathcal{R}_w$.

**Definition 3.2.** We denote by $\mathcal{P}$ the operator consisting in taking the primitive vashing at 0 and $\mathcal{L} = \mathrm{Id} - \mathcal{P}$. In what follows we shall consider functions of several variables, hence if there is an ambiguity on the variable, we shall note $\mathcal{P}_{x_i}$ and $\mathcal{L}_{x_i}$ when we apply these operators with respect to the variable $x_i$.

We shall use the following lemma in order to show that Friedman-Ramanujan functions are stable under convolution.

**Lemma 3.3 (Characterization of Friedman-Ramanujan functions, [AM25]).** $f \in \mathcal{F}^m$ (respectively in $\mathcal{F}_w^m$) if and only if $\mathcal{L}^m f \in \mathcal{R}$ (respectively $\mathcal{R}_w$).
We will also need to define a new class of functions similarly to the Friedman-Ramanujan functions, but with polynomial growth:

**Definition 3.4.** A function $f$ is said to belong to the class $SF$ (respectively $SF_w$) if $\ell \mapsto f(\ell)e^\ell$ is a Friedman-Ramanujan function (respectively a weak Friedman-Ramanujan function). In other words, $f$ belongs to $SF$ if there exist a polynomial $p$ and constants $c, c_1 > 0$ such that:

$$ \forall \ell > 0, |f(\ell) - p(\ell)| \le c_1(1 + \ell)^c e^{-\frac{\ell}{2}}, $$

and $f$ belongs to $SF_w$ if there exist a polynomial $p$ and constants $c, c_1 > 0$ such that:

$$ \forall n \ge 1, \int_0^n |f(\ell) - p(\ell)| e^\ell d\ell \le c_1 (1+n)^c e^{\frac{n}{2}}. \quad (6) $$

For $m \ge 0$, we also define $SF^m = e^{-\ell}SF^{m+1}$ and $SF_w^m = e^{-\ell}SF_w^{m+1}$.

In order to show that a function belongs to $SF$, we shall multiply it by $e^\ell$ and use Lemma 3.3.

<figure><img src="image_3.png" /></figure>

**3.2. Pseudo-convolution.** The pseudo-convolution provides a tool that generalizes the usual convolution of functions on $\mathbb{R}^n$ when disintegrating along level sets of non-linear functions.

**Definition 3.5 ([AM25]).** Let $\varphi : \mathbb{R}^n \to \mathbb{R}$ be a measurable and locally bounded function and $h$ a measurable function defined on the support of $\varphi$ such that for every compact subset $K \subset \mathbb{R}$, the set $h^{-1}(K) \cap \{\varphi \neq 0\}$ has finite measure.

Let $f_1, \dots, f_n \in L_{\text{loc}}^1(\mathbb{R})$. The $(h, \varphi)$-convolution of $f_1, \dots, f_n$ is the pushforward measure $\mu$ of $\varphi \times \bigotimes_{i=1}^n f_i dx_i$ by the map $h$, i.e. for every test function $g$:

$$ \int_{\mathbb{R}} g d\mu = \int_{\mathbb{R}^n} g(h(x_1, \dots, x_n)) \varphi(x_1, \dots, x_n) f_1(x_1) \cdots f_n(x_n) dx_1 \cdots dx_n. $$

When the measure $\mu$ admits a density with respect to the Lebesgue measure, we will denote for almost every $\ell \in \mathbb{R}$:

$$ \frac{d\mu}{d\ell}(\ell) = f_1 \star \cdots \star f_n(\ell) \Big|_{\varphi}^h. $$

Anantharaman and Monk gave in [AM25] a sufficient condition for the existence of $f_1 \star \cdots \star f_n(\ell)$ $\Big|_{\varphi}^h$.

**Definition 3.6 ([AM25]).** Let $h : \mathbb{R}_{\ge 0}^n \to \mathbb{R}$ be a proper map. We say that $h$ satisfies Assumption (H$_n$) if $h$ is $\mathscr{C}^1$ and if for any $1 \le i \le n$ and for any $(x_j)_{j \neq i} \in \mathbb{R}_{>0}^{n-1}$, the function:

$$ h_i = h_i^{(x_j)_{j \neq i}} : x_i \mapsto h(x_1, \dots, x_n) $$

is a $\mathscr{C}^1$-diffeomorphism from $\mathbb{R}_{>0}$ onto its image. Its inverse will then be denoted by:

$$ \ell \mapsto h_i^{-1}(\ell, (x_j)_{j \neq i}). $$

*Remark.* We can notice that, unlike in [AM25], we do not require the derivative $\frac{\partial h_i^{-1}}{\partial \ell}$ to be bounded on its domain of definition. This will have no impact on the existence almost everywhere of the pseudo-convolution, however this will have some consequences on its regularity.
**Proposition 3.7** ([AM25]). Under Assumption (H_n), the density $f_1 \star \cdots \star f_n|_{\varphi}^h$ exists, and for any $1 \le i \le n$ and any $\ell$ belonging in the image of h, the function $(x_j)_{j \ne i} \mapsto h_i^{-1}(\ell, (x_j)_{j \ne i})$ is defined on $U_{\ell}^i = \{(x_j)_{j \ne i} \in \mathbf{R}_{>0}^{n-1} | h_i^{(x_j)_{j \ne i}}(0) < \ell\}$ if h is an increasing function of $x_i$ and $U_{\ell}^i = \{(x_j)_{j \ne i} \in \mathbf{R}_{>0}^{n-1} | \lim_{x_i \to \infty} h_i^{(x_j)_{j \ne i}}(x_i) < \ell\}$ otherwise, and we have for almost every $\ell \in \mathbf{R}$:

$$f_1 \star \cdots \star f_n(\ell)|_{\varphi}^h = \int_{U_{\ell}^i} f_i(h_i^{-1}(\ell, (x_j)_{j \ne i})) \prod_{j \ne i} f_j(x_j) \times \varphi(x_1, \dots, x_{i-1}, h_i^{-1}(\ell, (x_j)_{j \ne i}), x_{i+1}, \dots, x_n) \left| \frac{\partial h_i^{-1}}{\partial \ell}(\ell, (x_j)_{j \ne i}) \right| \prod_{j \ne i} dx_j.$$

From now on, if we integrate a function $f \in L_{\text{loc}}^1(\mathbf{R}^n)$ on a set defined by an implicit relation $\ell = h(x_1, \dots, x_n)$ such that $h$ satisfies Assumption (H_n), we shall denote by:

$$\int_{\ell=h(x_1, \dots, x_n)} f(x_1, \dots, x_n) \frac{dx_1 \dots dx_n}{d\ell}$$

any of the equivalent expressions of:

$$\int_{U_{\ell}^{i}} f(x_1, \dots, x_{i-1}, h_{i}^{-1}(\ell, (x_j)_{j \neq i}), x_{i+1}, \dots, x_n) \left| \frac{\partial h_{i}^{-1}}{\partial \ell}(\ell, (x_j)_{j \neq i}) \right| \prod_{j \neq i} dx_j.$$

In particular we have for almost every $\ell \in \mathbf{R}$:

$$f_1 \star \cdots \star f_n(\ell)|_{\varphi}^h = \int_{\ell=h(x_1, \dots, x_n)} f_1(x_1) \cdots f_n(x_n) \varphi(x_1, \dots, x_n) \frac{dx_1 \cdots dx_n}{d\ell}.$$

### 3.3. Stability of Friedman-Ramanujan functions under pseudo-convolution.

**3.3.1. Computations in one variable.** We use the same notations as in [AM25]. Let $h$ be a diffeomorphism from $(0, +\infty)$ to $(h_{\text{inf}}, +\infty)$ and denote by $h \mapsto x(h)$ its inverse. For any function $f$, let us note:

$$R_x f(x) = \int_0^x f(t)(h'(t) - 1)dt.$$

Given a function $\varphi$, let:

$$T: f \mapsto (\varphi f) \circ x.$$

**Lemma 3.8.** Let $x_0 \ge 0$ and $\psi: \mathbf{R}_{\ge 0} \to \mathbf{R}$ be a smooth function. Denote $\varphi = \mathbf{1}_{(x_0, +\infty)}\psi$. For any continuous function $f: \mathbf{R} \to \mathbf{R}$ we have:

(i) $\mathcal{P}_h[f \circ x] = (\mathcal{P}_x f) \circ x + (R_x f) \circ x$;

(ii) $\mathcal{P}_x[\varphi f] = \varphi \mathcal{P}_x f - \mathcal{P}_x[\mathbf{1}_{(x_0, +\infty)}\psi' \mathcal{P}_x f] - \psi(x_0) \mathcal{P}_x f(x_0) \mathbf{1}_{(x_0, +\infty)}$.

*Proof.* (i) See [AM25].

(ii) We have:

$$\forall x \ge 0, \mathcal{P}_x[\varphi f](x) = \int_0^x \mathbf{1}_{(x_0, +\infty)}(t) \psi(t) f(t) dt = \mathbf{1}_{(x_0, +\infty)}(x) (\mathcal{P}_x[\psi f](x) - \mathcal{P}_x[\psi f](x_0)).$$

Moreover, integrating by parts:

$$\mathcal{P}_x[\psi f] = \psi \mathcal{P}_x f - \mathcal{P}_x[\psi' \mathcal{P}_x f].$$

It follows that:

$$\begin{align*}
\mathcal{P}_x[\varphi f] &= \mathbf{1}_{(x_0, +\infty)} \left( \psi \mathcal{P}_x f - \mathcal{P}_x[\psi' \mathcal{P}_x f] - \psi(x_0) \mathcal{P}_x f(x_0) + \mathcal{P}_x[\psi' \mathcal{P}_x f](x_0) \right) \\
&= \varphi \mathcal{P}_x f - \mathcal{P}_x[\mathbf{1}_{(x_0, +\infty)} \psi' \mathcal{P}_x f] - \psi(x_0) \mathcal{P}_x f(x_0) \mathbf{1}_{(x_0, +\infty)}
\end{align*}$$

□
Using this lemma we obtain:

**Lemma 3.9.** With the same notations as above:

$$[\mathcal{L}, T]f = -(\varphi R_x f) \circ x + \mathcal{P}_h[\partial_h(\psi \circ x)\mathcal{P}_h[f \circ x]1_{(x_0, +\infty)} \circ x] + \psi(x_0)\mathcal{P}_h[f \circ x](h(x_0))1_{(x_0, +\infty)} \circ x.$$

It follows:

**Proposition 3.10.** Let $K \ge 0$ be an integer. We have:

$$\mathcal{L}_h^K [(\varphi f) \circ x \frac{\partial x}{\partial h}] = (\varphi \mathcal{L}_x^K f) \circ x + \mathcal{L}_h^K [(\varphi f) \circ x (\frac{\partial x}{\partial h} - 1)] \quad (7)$$

$$+ \sum_{m=0}^{K-1} \mathcal{L}_h^{K-1-m} \mathcal{P}_h [1_{(x_0, +\infty)} \circ x \partial_h (\psi \circ x) \mathcal{P}_h [\mathcal{L}_x^m f \circ x]] \quad (8)$$

$$- \sum_{m=0}^{K-1-m} \mathcal{L}_h^{K-1-m} [(\varphi R_x \mathcal{L}_x^m f) \circ x] \quad (9)$$

$$+ \sum_{m=0}^{K-1} \mathcal{L}_h^{K-1-m} [\psi(x_0) 1_{(x_0, +\infty)} \circ x \mathcal{P}_h [\mathcal{L}_x^m f \circ x](h(x_0))] \quad (10)$$

*Proof.* As in [AM25], this identity comes from the fact that:

$$[\mathcal{L}^K, T] = \sum_{m=0}^{K-1} \mathcal{L}^{K-1-m}[\mathcal{L}, T]\mathcal{L}^m.$$

□

**3.3.2. Computations in several variables.** Let us introduce some notations. First, as in [AM25], let us note:

$$F_j^K(x_1, \dots, x_n) = \frac{\partial h}{\partial x_j}(x_1, \dots, x_n) \mathcal{L}_{x_j}^K f_j(x_j) \quad (11)$$

$$G_j(x_1, \dots, x_n) = \left(1 - \frac{\partial h}{\partial x_j}(x_1, \dots, x_n)\right) f_j(x_j) \quad (12)$$

$$H_j^m(x_1, \dots, x_n) = - R_{x_j} [\mathcal{L}_{x_j}^m f_j](x_1, \dots, x_n) \frac{\partial h}{\partial x_j}(x_1, \dots, x_n) \quad (13)$$

$$I_j^m(x_1, \dots, x_n) = \mathcal{P}_{x_j} \left[ \mathcal{L}_{x_j}^m [f_j] \frac{\partial h}{\partial x_j} \right] (x_1, \dots, x_n). \quad (14)$$

Let us also denote:

$$C^n = \{0 \le x_1 \le \dots \le x_n\}.$$

If $f$ is a function $\mathbf{R}^n \to \mathbf{C}$ and $1 \le i \le n$, let us note:

$$f^i(x_1, \dots, x_i) = f(x_1, \dots, x_i, \dots, x_i).$$

We finally use the bold font notation $\mathbf{x}_i$ to denote the vector $(x_1, \dots, x_i)$ and $d\mathbf{x}_i = dx_1 \cdots dx_i$.

With these notations in mind we are now able to prove the following:

**Theorem 3.11.** Let $K \ge 0$ be an integer, $\psi$ be a smooth function and $\varphi = 1_{C^n} \psi$. We assume that $h$ is a function satisfying the hypothesis $(\mathbf{H}_n)$ and such that for every $\mathbf{x}_{n-1} \in C^{n-1}$, $x_n \mapsto h(\mathbf{x}_{n-1}, x_n)$ is an
increasing function and that $h^{n-1}$ satisfies the hypothesis $(\mathbf{H}_{n-1})$. We have:

$$ \mathcal{L}_{\ell}^{K} [f_1 \star \cdots \star f_n(\ell)]_{\varphi}^{h} = \int_{\ell=h(\mathbf{x}_n)} f_1(x_1) \cdots f_{n-1}(x_{n-1}) F_n^K(\mathbf{x}_n) \varphi(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \quad (15) $$

$$ + \mathcal{L}_{\ell}^{K} \left[ \int_{\ell=h(\mathbf{x}_n)} f_1(x_1) \cdots f_{n-1}(x_{n-1}) G_n(\mathbf{x}_n) \varphi(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \right] \quad (16) $$

$$ - \sum_{m=0}^{K-1} \mathcal{L}_{\ell}^{K-1-m} \left[ \int_{\ell=h(\mathbf{x}_n)} f_1(x_1) \cdots f_{n-1}(x_{n-1}) H_n^m(\mathbf{x}_n) \varphi(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \right] \quad (17) $$

$$ + \sum_{m=0}^{K-1} \mathcal{L}_{\ell}^{K-1-m} \mathcal{P}_{\ell} \left[ \int_{\ell=h(\mathbf{x}_n)} f_1(x_1) \cdots f_{n-1}(x_{n-1}) I_n^m(\mathbf{x}_n) \mathbf{1}_{C^n}(\mathbf{x}_n) \frac{\partial \psi}{\partial x_n}(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \right] \quad (18) $$

$$ + \sum_{m=0}^{K-1} \mathcal{L}_{\ell}^{K-1-m} \mathcal{P}_{\ell} \left[ \int_{\ell=h^{n-1}(\mathbf{x}_{n-1})} f_1(x_1) \cdots f_{n-1}(x_{n-1}) (\varphi I_n^m)^{n-1}(\mathbf{x}_{n-1}) \frac{d\mathbf{x}_{n-1}}{d\ell} \right]. \quad (19) $$

*Proof.* As in [AM25], we split the integral

$$ f_1 \star \cdots \star f_n(\ell) \Big|_{\varphi}^{h} = \int \prod_{i=1}^{n-1} f_i(x_i) f_n(h_n^{-1}(\ell, \mathbf{x}_{n-1})) \varphi(\mathbf{x}_{n-1}, h_n^{-1}(\ell, \mathbf{x}_{n-1})) \frac{\partial h_n^{-1}}{\partial \ell}(\ell, \mathbf{x}_{n-1}) d\mathbf{x}_{n-1} $$

in two:

$$ f_1 \star \cdots \star f_n(\ell) \Big|_{\varphi}^{h} = \int \prod_{i=1}^{n-1} f_i(x_i) f_n(h_n^{-1}(\ell, \mathbf{x}_{n-1})) \varphi(\mathbf{x}_{n-1}, h_n^{-1}(\ell, \mathbf{x}_{n-1})) d\mathbf{x}_{n-1} \quad (20) $$

$$ + \int \prod_{i=1}^{n-1} f_i(x_i) f_n(h_n^{-1}(\ell, \mathbf{x}_{n-1})) \varphi(\mathbf{x}_{n-1}, h_n^{-1}(\ell, \mathbf{x}_{n-1})) \left( \frac{\partial h_n^{-1}}{\partial \ell}(\ell, \mathbf{x}_{n-1}) - 1 \right) d\mathbf{x}_{n-1}. \quad (21) $$

After rewriting

$$ \frac{\partial h_n^{-1}}{\partial \ell}(\ell, \mathbf{x}_{n-1}) - 1 = \frac{\partial h_n^{-1}}{\partial \ell}(\ell, \mathbf{x}_{n-1}) \left( 1 - \frac{\partial h}{\partial x_n}(\mathbf{x}_{n-1}, h_n^{-1}(\ell, \mathbf{x}_{n-1})) \right), $$

the term (21) gives the term (16).

The terms (15), (17), (18) and (19) come from applying Proposition 3.10 to (20). The computations for the terms (15), (17) and (18) are the same as in [AM25]. Let us explain how we obtain the term (19).

Using the term (10), we have to compute for $0 \le m \le K - 1$:

$$ \int \varphi^{n-1}(\mathbf{x}_{n-1}) \prod_{i=1}^{n-1} f_i(x_i) (I_n^m)^{n-1}(\mathbf{x}_{n-1}) \mathbf{1}_{\{x_{n-1} \le h_n^{-1}(\ell, \mathbf{x}_{n-1})\}} d\mathbf{x}_{n-1}. \quad (22) $$

Since $h$ is an increasing function of its last variable, we have $\mathbf{1}_{\{x_{n-1} \le h_n^{-1}(\ell, \mathbf{x}_{n-1})\}} = \mathbf{1}_{\{h^{n-1}(\mathbf{x}_{n-1}) \le \ell\}}$, hence after disintegrating along the level sets of $h^{n-1}$ the term (22) gives:

$$ \int_0^\ell \left( \int_{t=h^{n-1}(\mathbf{x}_{n-1})} \varphi^{n-1}(\mathbf{x}_{n-1}) \prod_{i=1}^{n-1} f_i(x_i) (I_n^m)^{n-1}(\mathbf{x}_{n-1}) \frac{d\mathbf{x}_{n-1}}{dt} \right) dt \\ = \mathcal{P}_\ell \left[ \int_{\ell=h^{n-1}(\mathbf{x}_{n-1})} f_1(x_1) \cdots f_{n-1}(x_{n-1}) (\varphi I_n^m)^{n-1}(\mathbf{x}_{n-1}) \frac{d\mathbf{x}_{n-1}}{d\ell} \right], $$

which in turn gives the term (19).

□
**Remark.** • Anantharaman and Monk proved a similar result when $\varphi$ is smooth in [AM25]. For a smooth function $\varphi$, the term (19) does not appear, but since in our computations $\varphi$ will not be continuous, a Dirac mass concentrated on the hyperplane $\{x_n = x_{n-1}\}$ will appear. This is the reason why, when $\varphi = 1_{C^n} \psi$, we shall denote:

$$ \int_{\ell=h(\mathbf{x}_n)} f_1(x_1) \cdots f_{n-1}(x_{n-1}) I_n^m(\mathbf{x}_n) \frac{\partial \varphi}{\partial x_n}(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \\ = \int_{\ell=h(\mathbf{x}_n)} f_1(x_1) \cdots f_{n-1}(x_{n-1}) I_n^m(\mathbf{x}_n) 1_{C^n}(\mathbf{x}_n) \frac{\partial \psi}{\partial x_n}(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \\ + \int_{\ell=h^{n-1}(\mathbf{x}_{n-1})} f_1(x_1) \cdots f_{n-1}(x_{n-1}) (\varphi I_n^m)^{n-1}(\mathbf{x}_{n-1}) \frac{d\mathbf{x}_{n-1}}{d\ell}. $$

With this notation, Theorem 3.11 can be rewritten as in [AM25]:

$$ \mathcal{L}_{\ell}^{K} [f_1 \star \cdots \star f_n(\ell)]_{\varphi}^{h} = \int_{\ell=h(\mathbf{x}_n)} f_1(x_1) \cdots f_{n-1}(x_{n-1}) F_n^K(\mathbf{x}_n) \varphi(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \\ + \mathcal{L}_{\ell}^{K} \left[ \int_{\ell=h(\mathbf{x}_n)} f_1(x_1) \cdots f_{n-1}(x_{n-1}) G_n(\mathbf{x}_n) \varphi(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \right] \quad (23) \\ - \sum_{m=0}^{K-1} \mathcal{L}_{\ell}^{K-1-m} \left[ \int_{\ell=h(\mathbf{x}_n)} f_1(x_1) \cdots f_{n-1}(x_{n-1}) H_n^m(\mathbf{x}_n) \varphi(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \right] \\ + \sum_{m=0}^{K-1} \mathcal{L}_{\ell}^{K-1-m} \mathcal{P}_{\ell} \left[ \int_{\ell=h(\mathbf{x}_n)} f_1(x_1) \cdots f_{n-1}(x_{n-1}) I_n^m(\mathbf{x}_n) \frac{\partial \varphi}{\partial x_n}(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \right]. $$

• The reason why we used non-strict inequalities to define $C^n$ is because if not, we would have $\varphi^{n-1} = 0$. However the identity (23) holds if $C^n$ is defined with any mix of strict and non-strict inequalities as long as $\frac{\partial \varphi}{\partial x_n}$ is a derivative taken in the sense of distributions. Thus in what follows, when considering cones defined in a similar manner to $C^n$, we won't really care about the fact that the inequalities we handle are actually strict ones or not.

**Theorem 3.12.** Let $h: \mathbf{R}_{>0}^n \to \mathbf{R}_{>0}$ be a smooth function such that:

$$ \forall \mathbf{x}_n \in \mathbf{R}_{>0}^n, h(\mathbf{x}_n) = x_n + \psi(\mathbf{x}_n), $$

and we assume that for any $1 \le i \le n$ there exists a constant $c_i > 0$ such that:

$$ \forall \mathbf{x}_i \in C^i, \left| \frac{\partial \psi^i}{\partial x_i}(\mathbf{x}_i) \right| \le c_i e^{-\frac{x_i}{2}}. $$

Finally we assume that $h^i$ satisfies the hypothesis $(\mathbf{H}_i)$ and is an increasing function of $x_i$ for any $1 \le i \le n$, and that there exists $A > \ell_0$ such that the diffeomorphism $h^1: x \mapsto h(x, \dots, x)$ satisfies the following estimates:

$$ \frac{d(h^1)^{-1}}{d\ell}(\ell) = \begin{cases} O\left(\frac{1}{\sqrt{\ell - \ell_0}}\right) & \text{on } (\ell_0, A) \\ 1 + O(e^{-\frac{\ell}{2}}) & \text{on } (A, +\infty) \end{cases} \quad (24) $$

Let $\varphi = 1_{C^n} e^\psi$ and $p_1 \in S\mathcal{F}^{K_1}, \dots, p_n \in S\mathcal{F}^{K_n}$. Then:

$$ p_1 \star \cdots \star p_{n-1} \star (e^{x_n} p_n) \Big|_{\varphi}^{h} \in \mathcal{F}_{w}^{K_1 + \cdots + K_n + n}. $$

**Proof.** We prove this result by induction on $n$. If $n=1$ then $h = h^1$ is a diffeomorphism and we have:

$$ \int_{\ell=h(x)} p_1(x) e^x \varphi(x) \frac{dx}{d\ell} = e^\ell p_1(h^{-1}(\ell)) \frac{dh^{-1}}{d\ell}(\ell). $$
By using the estimate (24), $(h^1)^{-1}$ is bounded on $(\ell_0, A)$, and there exists a constant $\ell_1$ such that
$h^{-1}(\ell) = \ell + \ell_1 + O(e^{-\frac{\ell}{2}})$ on $(A, +\infty)$. It follows that:

$$
\ell \mapsto \int_{\ell=h(x)} p_1(x) e^x \varphi(x) \frac{dx}{d\ell} \in \mathcal{F}_w^{K_1+1}.
$$

Now let us assume that the result is true for $n-1$ variables. Denote:

$$
Q(\ell) = p_1 \star \cdots \star p_{n-1} \star (p_n(x_n) e^{x_n})(\ell) \Big|_{\varphi}^{h}.
$$

We apply Theorem 3.11 with respect to the variable $x_n$. Let $K \ge 0$, we have:

$$
\mathcal{L}_{\ell}^{K} Q(\ell) = \int_{\ell=h(\mathbf{x}_n)} p_1(x_1) \cdots p_{n-1}(x_{n-1}) F_n^K(\mathbf{x}_n) \varphi(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \\
+ \mathcal{L}_{\ell}^{K} \left[ \int_{\ell=h(\mathbf{x}_n)} p_1(x_1) \cdots p_{n-1}(x_{n-1}) G_n(\mathbf{x}_n) \varphi(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \right] \\
- \sum_{m=0}^{K-1} \mathcal{L}_{\ell}^{K-1-m} \left[ \int_{\ell=h(\mathbf{x}_n)} p_1(x_1) \cdots p_{n-1}(x_{n-1}) H_n^m(\mathbf{x}_n) \varphi(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \right] \\
+ \sum_{m=0}^{K-1} \mathcal{L}_{\ell}^{K-1-m} \mathcal{P}_{\ell} \left[ \int_{\ell=h(\mathbf{x}_n)} p_1(x_1) \cdots p_{n-1}(x_{n-1}) I_n^m(\mathbf{x}_n) \frac{\partial \varphi}{\partial x_n}(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \right].
$$

Since the definition of $\varphi$ involves an indicator function, a Dirac mass will appear in its derivative. Indeed:

$$
\int_{\ell=h(\mathbf{x}_n)} p_1(x_1) \cdots p_{n-1}(x_{n-1}) I_n^m(\mathbf{x}_n) \frac{\partial \varphi}{\partial x_n}(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \\
= \int_{\ell=h(\mathbf{x}_n)} p_1(x_1) \cdots p_{n-1}(x_{n-1}) I_n^m(\mathbf{x}_n) \frac{\partial \psi}{\partial x_n}(\mathbf{x}_n) \varphi(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \\
+ \int_{\ell=h^{n-1}(\mathbf{x}_{n-1})} p_1(x_1) \cdots p_{n-1}(x_{n-1}) (I_n^m \varphi)^{n-1}(\mathbf{x}_{n-1}) \frac{d\mathbf{x}_{n-1}}{d\ell}.
$$

Denote:

$$
\Theta_m(\ell) = \int_{\ell=h^{n-1}(\mathbf{x}_{n-1})} p_1(x_1) \cdots p_{n-1}(x_{n-1}) (I_n^m \varphi)^{n-1}(\mathbf{x}_{n-1}) \frac{d\mathbf{x}_{n-1}}{d\ell}.
$$

With these notations, $\mathcal{L}_{\ell}^{K} Q(\ell)$ is a linear combination of $\mathcal{L}_{\ell}^{K-1-m} \mathcal{P}_{\ell} \Theta_m(\ell)$ and of (images under powers of $\mathcal{L}$ of) terms of the form:

$$
\int_{\ell=h(\mathbf{x}_n)} p_1(x_1) \cdots p_{n-1}(x_{n-1}) \Phi(\mathbf{x}_n) \varphi(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell},
$$

where $\Phi$ is either $F_n^K$, $G_n$, $H_n^m$ or $\frac{\partial \psi}{\partial x_n} I_n^m$. By our assumption on $\varphi$ there exists a constant $C_0$ and an integer $N_0$ such that, as long as $K \ge K_1 + 1$:

$$
\forall \mathbf{x}_n \in C^n, |\Phi(\mathbf{x}_n)| \le C_0(1 + x_n)^{N_0} e^{\frac{x_n}{2}}.
$$

Under our assumptions on $h$ and $\psi$, there exists a constant $\kappa \in \mathbf{R}$ such that $x_n \le h(\mathbf{x}_n) + \kappa$ on $C^n$. Hence, taking possibly greater $C_0$ and $N_0$, if $a \ge 1$:

$$
\left| \int_0^a \left( \int_{\ell=h(\mathbf{x}_n)} p_1(x_1) \cdots p_{n-1}(x_{n-1}) \Phi(\mathbf{x}_n) \varphi(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \right) d\ell \right| \le C_0 (1+a)^{N_0} e^{\frac{a}{2}} \int 1_{C^n \cap \{h(\mathbf{x}_n) \le a\}} d\mathbf{x}_n.
$$

Since $x_n \le h(\mathbf{x}_n) + \kappa$, there exists a constant $C_1 > 0$ such that:

$$
\int 1_{C^n \cap \{h(\mathbf{x}_n) \le a\}} d\mathbf{x}_n \le C_1 (1 + a)^n.
$$
Hence:

$$ \ell \mapsto \int_{\ell=h(\mathbf{x}_n)} p_1(x_1) \cdots p_{n-1}(x_{n-1}) \Phi(\mathbf{x}_n) \varphi(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \in \mathcal{R}_w. \quad (25) $$

We now have to deal with $\Theta_m$ in order to conclude.

By definition we have:

$$ I_n^m(\mathbf{x}_n) = \mathcal{P}_{x_n} \mathcal{L}_{x_n}^m [p_n(x_n) e^{x_n}] + \mathcal{P}_{x_n} \left[ \mathcal{L}_{x_n}^m [p_n(x_n) e^{x_n}] \frac{\partial \psi}{\partial x_n}(\mathbf{x}_n) \right]. $$

Let us note:

$$ \begin{aligned} f(x_n) &= \mathcal{P}_{x_n} \mathcal{L}_{x_n}^m [p_n(x_n) e^{x_n}] \\ r(\mathbf{x}_n) &= \mathcal{P}_{x_n} \left[ \mathcal{L}_{x_n}^m [p_n(x_n) e^{x_n}] \frac{\partial \psi}{\partial x_n}(\mathbf{x}_n) \right]. \end{aligned} $$

On the one hand, by our assumptions on $\psi$ we obtain a constant $C_2$ and an integer $N_1$ such that:

$$ \forall \mathbf{x}_n \in C^n, |r(\mathbf{x}_n)| \le C_2(1 + x_n)^{N_1} e^{\frac{x_n}{2}}. $$

Thus, as for the estimate (25), we can conclude that:

$$ \ell \mapsto \int_{\ell=h^{n-1}(\mathbf{x}_{n-1})} p_1(x_1) \cdots p_{n-1}(x_{n-1}) r^{n-1}(\mathbf{x}_{n-1}) \varphi^{n-1}(\mathbf{x}_{n-1}) \frac{d\mathbf{x}_{n-1}}{d\ell} \in \mathcal{R}_w. $$

On the other hand $f \in \mathcal{F}^{K_n-m+1}$ and $p_{n-1}f \in \mathcal{F}^{K_{n-1}+K_n-m+1}$. Using the induction hypothesis on $p_1 \star \cdots \star p_{n-2} \star (p_{n-1}f)(\ell)|_{\varphi^{n-1}}^{h^{n-1}}$, we can conclude that:

$$ \Theta_m \in \mathcal{F}_w^{K_1+\cdots+K_n+n-1-m}. $$

Hence $\mathcal{L}^{K-1-m}\Theta_m \in \mathcal{F}_w^{K_1+\cdots+K_n+n-K}$ and $Q \in \mathcal{F}_w^{K_1+\cdots+K_n+n}$. $\square$

It immediately follows:

**Corollary 3.13.** Under the same assumptions:

$$ p_1 \star \cdots \star p_n \Big|_{1 \le c_n}^h \in S \mathcal{F}_w^{K_1+\cdots+K_n+n-1}. $$

*Remark.* The two previous results hold if $C^n$ is defined with any mix of strict and non-strict inequalities. However the induction does not hold anymore if we consider other cones than $C^n$.

**3.4. Regularity of the convolution.** In this section we shall discuss the regularity of functions defined the following way:

$$ F: \ell \mapsto \int_{\ell=h(\mathbf{x}_n)} f(\mathbf{x}_n) \mathbf{1}_{\mathbf{R}_{\ge 0}^n}(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell}. \quad (26) $$

Here we assume that $n \ge 2$. Such a function is well defined and belongs to $L_{loc}^1(\mathbf{R})$ as soon as $f \in L_{loc}^1(\mathbf{R}_{\ge 0})$. Moreover $\text{supp}(F) \subset h(\mathbf{R}_{\ge 0}^n)$.

We denote by $W_{loc}^{1,1}(\mathbf{R}_{\ge 0}^n)$ the space of functions which have derivatives in $L_{loc}^1(\mathbf{R})$ in the sense of distributions. We also denote, for $1 \le i \le n$:

$$ D_i f = \frac{\partial}{\partial x_i} \left( f \middle/ \frac{\partial h}{\partial x_i} \right). \quad (27) $$

**Definition 3.14.** Let $h: \mathbf{R}_{\ge 0}^n \to \mathbf{R}$ be a proper map and $f \in L_{loc}^1(\mathbf{R}_{\ge 0}^n)$. We say that the couple $(h, f)$ satisfies Assumption $(H'_n)$ if:

(i) $h$ satisfies Assumption $(H_n)$ and for all $1 \le i \le n$, $\hat{x}_i \mapsto h^{\hat{x}_i}(0)$ satisfies Assumption $(H_{n-1})$;

(ii) $f \in W_{loc}^{1,1}(\mathbf{R}_{\ge 0}^n) \cap \mathcal{C}^0(\mathbf{R}_{\ge 0}^n)$;
(iii) for all $1 \le i \le n$, $f/\frac{\partial h}{\partial x_i}$ extends continuously to the hyperplane $\{x_i = 0\} \cap \mathbf{R}_{\ge 0}^n$ to a locally integrable function of the variable $(x_j)_{j \ne i} \in \mathbf{R}_{\ge 0}^{n-1}$ and $D_i f \in L_{\text{loc}}^1(\mathbf{R}_{\ge 0}^n)$.

**Lemma 3.15.** Let $F$ be defined as the integral (26). Under Assumption (H'_n), we have $F \in W_{\text{loc}}^{1,1}(\mathbf{R})$ and for almost every $\ell \in \mathbf{R}$:

$$F'(\ell) = \int_{\ell=h(\mathbf{x}_{n-1},0)} \frac{f(\mathbf{x}_{n-1},0)}{\frac{\partial h}{\partial x_n}(\mathbf{x}_{n-1},0)} \mathbf{1}_{\mathbf{R}_{\ge 0}^{n-1}}(\mathbf{x}_{n-1}) \frac{d\mathbf{x}_{n-1}}{d\ell} + \int_{\ell=h(\mathbf{x}_n)} D_n f(\mathbf{x}_n) \mathbf{1}_{\mathbf{R}_{\ge 0}^n}(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell}.$$

In particular $F$ is continuous.

*Proof.* Let us assume that $h$ is an increasing function of $x_n$ (if not, we substitute $h$ for $-h$) and $\varphi \in \mathcal{D}(\mathbf{R})$. According to Proposition 3.7, we have:

$$\begin{aligned}
\int_{\mathbf{R}} \varphi'(\ell) F(\ell) d\ell &= \int_{\mathbf{R}} \varphi'(\ell) \left( \int_{U_{\ell}^n} \frac{f(\mathbf{x}_{n-1}, h_n^{-1}(\ell, \mathbf{x}_{n-1}))}{\frac{\partial h}{\partial x_n}(\mathbf{x}_{n-1}, h_n^{-1}(\ell, \mathbf{x}_{n-1}))} d\mathbf{x}_{n-1} \right) d\ell \\
&= \int_{\mathbf{R}_{\ge 0}^n} \left( \int_{h(\mathbf{x}_{n-1},0)}^{\infty} \varphi'(\ell) \frac{f(\mathbf{x}_{n-1}, h_n^{-1}(\ell, \mathbf{x}_{n-1}))}{\frac{\partial h}{\partial x_n}(\mathbf{x}_{n-1}, h_n^{-1}(\ell, \mathbf{x}_{n-1}))} d\ell \right) d\mathbf{x}_{n-1} \\
&= - \int_{\mathbf{R}_{\ge 0}^{n-1}} \varphi(h(\mathbf{x}_{n-1},0)) \frac{f(\mathbf{x}_{n-1},0)}{\frac{\partial h}{\partial x_n}(\mathbf{x}_{n-1},0)} d\mathbf{x}_{n-1} \\
&\quad - \int_{\mathbf{R}_{\ge 0}^{n-1}} \left( \int_{h(\mathbf{x}_n,0)}^{\infty} \varphi(\ell) D_n f(\mathbf{x}_{n-1}, h_n^{-1}(\ell, \mathbf{x}_{n-1})) \frac{\partial h_n^{-1}}{\partial \ell}(\ell, \mathbf{x}_{n-1}) d\ell \right) d\mathbf{x}_{n-1}.
\end{aligned}$$

Hence by disintegrating along the level sets of $\mathbf{x}_{n-1} \mapsto h(\mathbf{x}_{n-1},0)$, we obtain:

$$\begin{aligned}
\int_{\mathbf{R}} \varphi'(\ell) F(\ell) d\ell = & - \int_{\mathbf{R}} \varphi(\ell) \left( \int_{\ell=h(\mathbf{x}_{n-1},0)} \frac{f(\mathbf{x}_{n-1},0)}{\frac{\partial}{\partial x_n}(\mathbf{x}_{n-1},0)} \mathbf{1}_{\mathbf{R}_{\ge 0}^{n-1}}(\mathbf{x}_{n-1}) \frac{d\mathbf{x}_{n-1}}{d\ell} \right) d\ell \\
& - \int_{\mathbf{R}} \varphi(\ell) \left( \int_{\ell=h(\mathbf{x}_n)} D_n f(\mathbf{x}_n) \mathbf{1}_{\mathbf{R}_{\ge 0}^n}(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \right) d\ell.
\end{aligned}$$

□

*Remark.* The conclusion of Lemma 3.15 holds by replacing $D_n f$ and $\frac{f(\mathbf{x}_{n-1},0)}{\frac{\partial h}{\partial x_n}(\mathbf{x}_{n-1},0)}$ by any of the $D_i f$'s and $\frac{f(x_1, \dots, x_{i-1}, 0, x_{i+1}, \dots, x_n)}{\frac{\partial h}{\partial x_i}(x_1, \dots, x_{i-1}, 0, x_{i+1}, \dots, x_n)}$'s respectively for $1 \le i \le n$.

### 4. APPLICATION TO THE INTEGRATION OF GEOMETRIC RANDOM VARIABLES

Let $\gamma$ be an eight-shaped curve. In what follows, we will use the same notations as in Section 2. Namely:

$$h : (x, y, z) \in \mathbf{R}_{>0}^3 \mapsto 2\operatorname{argcosh}\left(2\cosh\left(\frac{x}{2}\right)\cosh\left(\frac{y}{2}\right) + \cosh\left(\frac{z}{2}\right)\right).$$

Therefore we can write the function $V_{\mathcal{O}_{\gamma}}$, which verifies:

$$\mathbb{E}[F^{\gamma}] = \int_{0}^{\infty} F(\ell) V_{\mathcal{O}_{\gamma}}(\ell, L) \frac{d\ell}{V_{g,n}(L)},$$

as a linear combination of pseudo-convolutions of polynomials. Indeed, take for example the case of an eight-shaped curve of type ($\mathcal{P}_0$), we can write $\mathbb{E}[F^{\gamma}]$ as a linear combination of terms of the form:

$$\int_{\mathbf{R}_{>0}^3} F(h(x, y, z)) x^i y^j z^k dxdydz = \int_0^\infty F(\ell) \left(x^i \star y^j \star z^k(\ell)\right) \left|_{\mathbf{1}_{\mathbf{R}_{>0}^3}}^h \right. d\ell,$$

with $i + j + k \le 6g - 9 + 2n$ and $i, j, k \ge 1$.
**Lemma 4.1.** (i) Let $f_1, g_1: \mathbf{R}_{\ge 0}^3 \to \mathbf{R}$ be $\mathscr{C}^1$ functions such that:
$$ \forall x, y, z \ge 0, f_1(x, y, z) = xyzg_1(x, y, z) $$

Then $(h, f_1)$ satisfies Assumption $(\mathbf{H}'_3)$.

(ii) Let $f_2, g_2: \mathbf{R}_{\ge 0}^2 \to \mathbf{R}$ be $\mathscr{C}^1$ functions such that:
$$ \forall u, v \ge 0, f_2(u, v) = uvg_2(u, v), $$

and denote:

$$ h_1: (x, z) \mapsto h(x, L, z) $$

$$ \tilde{h}_1: (x, y) \mapsto h(x, y, L) $$

$$ h_2: (x, z) \mapsto h(x, x, z) $$

$$ \tilde{h}_2: (x, y) \mapsto h(x, y, x), $$

for some $L \ge 0$. Then $(h_i, f_2)$ and $(\tilde{h}_i, f_2)$ satisfy Assumption $(\mathbf{H}'_2)$ for $i \in \{1, 2\}$.

*Proof.* (i) Let us make the computations for the variable $z$ (the computations are very similar when considering the variables $x$ or $y$). We have:

$$ \frac{\partial h}{\partial z} = \frac{\sinh(\frac{z}{2})}{\sinh(\frac{h}{2})}, $$

hence $f_1/\partial h/\partial z$ extends continuously to the plane $\{z=0\}$ as the function:

$$ (x, y) \mapsto 2xy \sinh\left(\frac{h(x, y, 0)}{2}\right) g_1(x, y, 0), $$

which is locally integrable on $\mathbf{R}_{\ge 0}^2$.

We now need to check that $\tilde{D}_3 f_1 \in L_{loc}^1(\mathbf{R}_{\ge 0}^3)$. We have $D_3 f_1 \in L_{loc}^1(\mathbf{R}_{>0}^3)$, thus we need to check that $D_3 f_1$ is integrable on compact neighborhoods of the plane $\{z=0\}$. We have:

$$ D_3 f_1 = \frac{\sinh(\frac{h}{2})}{\sinh(\frac{z}{2})} \left( \frac{\partial f_1}{\partial z} - \frac{f_1}{2 \tanh(\frac{z}{2})} \right) + \frac{f_1}{2 \tanh(\frac{h}{2})}. \quad (28) $$

Since $f_1(x, y, z) = xyzg_1(x, y, z)$ for all $x, y, z \ge 0$, $D_3 f$ extends continuously on $\mathbf{R}_{\ge 0}^3$ and is therefore locally integrable.

(ii) The computations we made in (i) implie that $(h_1, f_2)$ and $(\tilde{h}_1, f_2)$ satisfy Assumption $(\mathbf{H}'_2)$. Let us check that $(h_2, f_2)$ satisfies Assumption $(\mathbf{H}'_2)$. $f_2/\partial h_2/\partial y$ extends continuously on $\{y=0\}$ to the function:

$$ y \mapsto \frac{2y \sinh(\frac{h_2(x, 0)}{2})}{1 + 2 \cosh(\frac{0}{2})} g_2(x, 0), $$

which is locally integrable on $\mathbf{R}_{\ge 0}$, and we have:

$$ D_2 f_2 = \frac{\sinh(\frac{h_2}{2})}{(1 + 2 \cosh(\frac{x}{2})) \sinh(\frac{y}{2})} \left( \frac{\partial f_2}{\partial y} - \frac{f_2}{2 \tanh(\frac{y}{2})} \right) + \frac{f_2}{2 \tanh(\frac{h_2}{2})}. $$

The same computation as the one we did for (28) enables us to conclude that $D_2 f_2 \in L_{loc}^1(\mathbf{R}_{\ge 0}^2)$. We can similarly extend $f_2/\partial h_2/\partial x$ on the line $\{x=0\}$ to a locally integrable function on $\mathbf{R}_{\ge 0}$ and that $D_1 f_2 \in L_{loc}^1(\mathbf{R}_{\ge 0}^2)$. Hence $(h_2, f_2)$ satisfies Assumption $(\mathbf{H}'_2)$.

The same computations as before also enable us to conclude that $(\tilde{h}_2, f_2)$ satisfies Assumption $(\mathbf{H}'_2)$. $\square$
**Corollary 4.2.** If $\gamma$ is of type $(\mathcal{P}_0), (\mathcal{P}_1), (\mathcal{P}'_1), (\mathcal{T}_0)$ or $(\mathcal{T}'_0)$, there exists a continuous function $V_{\mathcal{O}_\gamma}(\cdot, L):$ $\mathbf{R} \to \mathbf{R}$ such that for every test function $F: \mathbf{R} \to \mathbf{C}$:

$$ \mathbb{E}[F^{\gamma}] = \int_{\mathbf{R}} F(\ell) V_{\mathcal{O}_{\gamma}}(\ell, L) \frac{d\ell}{V_{g,n}(L)}. $$

Moreover:

(i) if $\gamma$ is of type $(\mathcal{P}_0), (\mathcal{T}_0)$ or $(\mathcal{T}'_0)$, then $\mathrm{supp}(V_{\mathcal{O}_\gamma}(\cdot, L)) \subset [2\mathrm{argcosh}(3), +\infty)$;

(ii) if $\gamma$ is of type $(\mathcal{P}_1)$, there exists $1 \le i \le n$ such that $\mathrm{supp}(V_{\mathcal{O}_\gamma}(\cdot, L)) \subset [h(0, L_i, 0), +\infty)$;

(iii) if $\gamma$ is of type $(\mathcal{P}'_1)$, there exists $1 \le i \le n$ such that $\mathrm{supp}(V_{\mathcal{O}_\gamma}(\cdot, L)) \subset [h(0, 0, L_i), +\infty)$.

For convenience in our computations, let us define:

$$ H: (x, y, z) \mapsto 2 \log \left( 2 \cosh \left( \frac{h(x, y, z)}{2} \right) \right). \qquad (29) $$

$H$ is easier to handle in terms of computations than $h$, and since $2\log(2\cosh(\ell/2)) = \ell + O(e^{-\ell})$, we have:

$$ x^i \star y^j \star z^k(\ell) \Big|_{\mathbb{I}_{\mathbf{R}^3_{>0}}}^h = (1 + O(e^{-\ell})) \left( x^i \star y^j \star z^k(\ell + O(e^{-\ell})) \right) \Big|_{\mathbb{I}_{\mathbf{R}^3_{>0}}}^H. \qquad (30) $$

Hence if we manage to prove that $x^i \star y^j \star z^k(\ell)\Big|_{\mathbb{I}_{\mathbf{R}^3_{>0}}}^H$ belongs to $\mathcal{SF}_w$, the same goes on for $x^i \star y^j \star z^k(\ell)\Big|_{\mathbb{I}_{\mathbf{R}^3_{>0}}}^h$.

### 4.1. Curves exactly filling a pair of pants.

**4.1.1. Generic case.** We suppose first that $\gamma$ is an eight-shaped curve of type $(\mathcal{P}_0)$. To deal with this case, we will exploit the fact that $H(x, y, z)$ is close to $\max\{x+y, z\}$ in a quantitative way. We shall split $\mathbf{R}^3_{>0}$ into several sectors in which we will be able to control the error term $H(x, y, z) - \max\{x+y, z\}$. After changing the variables in each of these sectors we will eventually be able to conclude.

• Let us first suppose that $x+y > z$. Under this assumption we can write:

$$ H(x, y, z) = x + y + 2 \log \left[ 1 + e^{-x} + e^{-y} + e^{-x-y} + e^{-\frac{x-y+z}{2}} + e^{-\frac{x-y-z}{2}} \right]. $$

We can notice that we can write $H$ in the following form:

$$ H(x, y, z) = \Lambda(x, y, z) + 2 \log \left[ 1 + \sum_{T} e^{-\frac{T(x, y, z)}{2}} \right]. \qquad (31) $$

Here $\Lambda$ is a linear form and the sum is taken over a finite family of linear forms on $\mathbf{R}^3$, such that both $\Lambda$ and the $T$'s have integer coefficients and are positive on $\mathbf{R}^3_{>0} \cap \{x+y > z\}$.

Now that we have written $H$ as a function of the form (31), let us check all the possible ways of ordering the linear forms involved in its expression:

$$ 0 < x + y - z < 2x < x + y < 2y < x + y + z < 2x + 2y \qquad (32) $$

$$ 0 < x + y - z < 2y < x + y < 2x < x + y + z < 2x + 2y \qquad (33) $$

$$ 0 < 2x < x + y - z < x + y < x + y + z < 2y < 2x + 2y \qquad (34) $$

$$ 0 < 2y < x + y - z < x + y < x + y + z < 2x < 2x + 2y. \qquad (35) $$

**Remark.** Recall that our results on pseudo-convolutions hold in cones defined with any mix of strict and non-strict inequalities, hence it does not really matter to have only strict inequalities in the orders (32) to (35).
Since $x$ and $y$ have symmetric roles, it is sufficient to focus only on the orders (32) and (34). Let us focus on (32) first, which is equivalent to:

$$0 < x + y - z < 2x < x + y.$$

Thus, substituting the variables $x$ for $s = x + y - z$, $y$ for $t = 2x$ and $z$ for $u = x + y$ defines a change of variables mapping the sector defined by (32) to the cone $C = \{0 < s < t < u\}$. The same change of variables maps the sector defined by (34) to the cone $C' = \{0 < t < s < u\}$, and since the new expression $\tilde{H}$ of $H$ in the variable $(s, t, u)$:

$$\tilde{H}(s, t, u) = u + 2 \log \left[ 1 + e^{-\frac{s}{2}} + e^{-\frac{t}{2}} + e^{\frac{s}{2}-u} + e^{\frac{t}{2}-u} + e^{-u} \right]$$

is symmetric with respect to $s$ and $t$, it is sufficient to focus only on the order (32).

With this change of variables, we can write $x^i \star y^j \star z^k(\ell)|_{\mathbb{1}_{\{0<x+y-z<2x<x+y\}}}^H$ as a linear combination of terms of the form $s^{i'} \star t^{j'} \star u^{k'}(\ell)|_{\mathbb{1}_C}^{\tilde{H}}$, with $i'+j'+k' = i+j+k$. Indeed, denoting $U = \{0 < x + y - z < 2x < x + y\}$ we have:

$$\begin{aligned}
x^i \star y^j \star z^k(\ell)|_{\mathbb{1}_U}^H &= \int_{\ell=H(x,y,z)} x^i y^j z^k \mathbb{1}_U(x,y,z) \frac{dx dy dz}{d\ell} \\
&= \frac{1}{2} \int_{\ell=\tilde{H}(s,t,u)} (\frac{t}{2})^i (u-\frac{t}{2})^j (u-s)^k \mathbb{1}_C(s,t,u) \frac{ds dt du}{d\ell} \\
&= \sum_{p+p'=j} \sum_{q+q'=k} \frac{(-1)^{p'+q'} j! k!}{2^{i+p'+1} p! p'! q! q'} s^{q'} \star t^{i+p'} \star u^{p+q}(\ell)|_{\mathbb{1}_C}^{\tilde{H}}
\end{aligned}$$

We can therefore use Theorem 3.12 to obtain the following:

**Lemma 4.3.** With the same notations as above:

$$\ell \mapsto s^{i'} \star t^{j'} \star u^{k'}(\ell)|_{\mathbb{1}_C}^H \in \mathcal{SF}_w^{i'+j'+k+2}.$$

As a direct consequence:

**Corollary 4.4.**

$$x^i \star y^j \star z^k(\ell)|_{\mathbb{1}_{\{x+y>z\}}}^H \in \mathcal{SF}_w^{i+j+k+2}.$$

* Let us now assume that $x + y < z$. We can then write:

$$H(x, y, z) = z + 2 \log \left[ 1 + e^{-z} + e^{\frac{x+y-z}{2}} + e^{\frac{x-y-z}{2}} + e^{\frac{-x+y-z}{2}} + e^{\frac{-x-y-z}{2}} \right].$$

There are two possible ways of ordering the linear forms involved in the expression of $H$:

$$0 < z - x - y < z - x + y < z < z + x - y < z + x + y < 2z \qquad (36)$$

$$0 < z - x - y < z + x - y < z < z - x + y < z + x + y < 2z. \qquad (37)$$

Once again the symmetry between the variables $x$ and $y$ enables us to focus only on the order (36). Substitute the variables $x$ for $s = z - x - y$ and $y$ for $t = z - x + y$ (and keep $z$ unchanged). Since the order (36) is equivalent to $0 < z - x - y < z - x + y < z$, the change of variable from $(x, y, z)$ to $(s, t, z)$ maps the sector defined by (36) to $\{0 < s < t < z\}$. After rewriting $H$ as a function of $(s, t, z)$, we can therefore apply Theorem 3.12 to conclude.

By putting together this partial conclusion with Corollary 4.4 we just proved the following:
**Lemma 4.5.** Let $\gamma$ be an eight-shaped curve of type ($\mathcal{P}_0$). There exist functions $\{V_{\mathcal{O}_\gamma}(\cdot, L)\}_{L \in \mathbb{R}_{\ge 0}^n} \subset \mathcal{S}\mathcal{F}_w^{6g-7+2n}$ such that for every test function $F: \mathbf{R} \to \mathbf{C}$:

$$ \mathbb{E}[F^{\gamma}] = \int_{0}^{\infty} F(\ell) V_{\mathcal{O}_{\gamma}}(\ell, L) \frac{d\ell}{V_{g,n}(L)}. $$

4.1.2. Curves close to the boundary. We now assume that some boundaries of the surface $S(\gamma)$ filled by $\gamma$ are non-essential curves.

* If $\gamma$ is of type ($\mathcal{P}_1$), we split $\mathbb{R}_{>0}^2$ into the half-planes $\{x > z\}$ and $\{x < z\}$. If $x > z$, we have:

$$ H(x, L_i, z) = x + 2 \log \left[ 2 \cosh \left( \frac{L_i}{2} \right) (1 + e^{-x}) + e^{\frac{z-x}{2}} + e^{-\frac{x+z}{2}} \right]. $$

There is only one way of ordering these linear forms:

$$ 0 < x - z < x < x + z < 2x. $$

Hence, substituting $x$ for $s = x - z$ and $y$ for $t = x$, we can use Theorem 3.12 to conclude. We can repeat the same process in the half plane $\{x < z\}$.

* If $\gamma$ is of type ($\mathcal{P}'_1$) we have:

$$ H(x, y, L_i) = x + y + 2 \log \left[ 1 + e^{-x} + e^{-y} + e^{-x-y} + 2 \cosh \left( \frac{L_i}{2} \right) e^{-\frac{x+y}{2}} \right]. $$

Thus there are two possible ways of ordering the linear forms involved in the expression of this function:

$$ 0 < 2x < x + y < 2y < 2x + 2y \quad (38) $$

$$ 0 < 2y < x + y < 2x < 2x + 2y. \quad (39) $$

The symmetry between $x$ and $y$ makes it enough to focus only on (38). After substituting $x$ for $s = 2x$ and $y$ for $t = x + y$, we can conclude by using Theorem 3.12.

* If $\gamma$ is of type ($\mathcal{P}_2$), we only need to use the initializing step of the induction in the proof of Theorem 3.12. Indeed, according to Corollary 2.5, for every test function $F: \mathbf{R} \to \mathbf{C}$ we have:

$$ \begin{aligned} \mathbb{E}[F^{\gamma}] &= \int_{0}^{\infty} F(h(L_i, L_j, z)) Q_{\gamma}^{g,n,L}(z) \frac{dz}{V_{g,n}(L)} \\ &= \int_{h(L_i, L_j, 0)}^{\infty} F(\ell) Q_{\gamma}^{g,n,L} \left( h_3^{-1}(\ell, L_i, L_j) \right) \frac{\partial h_3^{-1}}{\partial \ell}(\ell, L_i, L_j) \frac{d\ell}{V_{g,n}(L)}. \end{aligned} $$

Hence in this case we have:

$$ \forall \ell \ge 0, V_{\mathcal{O}_{\gamma}}(\ell, L) = Q_{\gamma}^{g,n,L} \left( h_3^{-1}(\ell, L_i, L_j) \right) \frac{\partial h_3^{-1}}{\partial \ell}(\ell, L_i, L_j) \mathbf{1}_{(h(L_i, L_j, 0), +\infty)}(\ell). \quad (40) $$

By definition of $h$ we can compute:

$$ h_3^{-1}(\ell, L_i, L_j) = 2 \operatorname{argcosh} \left( \cosh\left(\frac{\ell}{2}\right) - 2 \cosh\left(\frac{L_i}{2}\right) \cosh\left(\frac{L_j}{2}\right) \right), $$

and:

$$ \frac{\partial h_3^{-1}}{\partial \ell}(\ell, L_i, L_j) = \frac{\sinh(\frac{\ell}{2})}{\sqrt{\left(\cosh(\frac{\ell}{2}) - 2 \cosh(\frac{L_i}{2}) \cosh(\frac{L_j}{2})\right)^2 - 1}}. $$

* The same argument can if $\gamma$ is of type ($\mathcal{P}'_2$). Thus we have:

$$ \forall \ell \ge 0, V_{\mathcal{O}_{\gamma}}(\ell, L) = \frac{2 \cosh(\frac{L_i}{2}) \sinh(\frac{\ell}{2})}{\sqrt{\left(\cosh(\frac{\ell}{2}) - \cosh(\frac{L_j}{2})\right)^2 - 4 \cosh^2(\frac{L_i}{2})}} Q_{\gamma} \left( 2 \operatorname{argcosh} \left( \frac{\cosh(\frac{\ell}{2}) - \cosh(\frac{L_j}{2})}{2 \cosh(\frac{L_i}{2})} \right) \right). \quad (41) $$

Finally:
**Lemma 4.6.** The conclusion of Lemma 4.5 holds for any eight-shaped curve exactly filling a pair of pants.

**4.2. Curves almost filling a holed torus.** We still have to deal with curves $\gamma$ which almost fill a holed torus.

**4.2.1. A first generic case.** Let $\gamma$ be a curve of type $(\mathcal{T}_0)$. In this case, in the regime $2x > z$ we have:

$$H(x, x, z) = 2x + 2 \log \left[ 1 + 2e^{-x} + e^{-2x} + e^{\frac{\pi}{2}-x} + e^{-\frac{\pi}{2}-x} \right].$$

The linear forms appearing in the expression of this function can be ordered in only one way:

$$0 < 2x - z < 2x < 2x + z < 4x.$$

Thus when substituting $x$ for $s = 2x - z$ and $z$ for $t = 2x$, we can conclude by using Theorem 3.12.

In the regime $2x > z$ we rewrite:

$$\tilde{H}(x, z) = z + 2 \log \left[ 1 + 2e^{-\frac{\pi}{2}} + e^{-z} + e^{\frac{\pi}{2}-z} + e^{-\frac{\pi}{2}-z} \right].$$

Once again, there is only one way of ordering these linear forms:

$$0 < z - 2x < z < z + 2x < 2z,$$

which leads to a change of variables enabling us to use Theorem 3.12.

**Lemma 4.7.** The conclusion of Lemma 4.5 holds if $\gamma$ is of type $(\mathcal{T}_0)$.

**4.2.2. Issues with the usual change of variables.** We now focus on geodesics of type $(\mathcal{T}_0')$. The method we will use to deal with this case is a little different from the one we used in the previous cases, and is more similar to the method used by Anantharaman and Monk in [AM25]. Indeed, until now our method consisted in splitting $\mathbf{R}^3_{>0}$ (respectively $\mathbf{R}^2_{>0}$) into cones that are defined by ordering the linear forms involved in the expression of $H$. When taking the three smallest linear forms of a given order (respectively the two smallest linear forms), we obtained a linear isomorphism mapping the associated cone to $\{0 < s < t < u\}$ (respectively $\{0 < s < t\}$), and we took advantage of the behavior of the new function we obtained this way by using Theorem 3.12. Contrary to the previous cases, this method does not work for curves of type $(\mathcal{T}_0')$ because the different ways of ordering the linear forms involved in the expression of $H$ do not enable us to match the requirements for using Theorem 3.12. Let us explain why.

Let us note:

$$\tilde{H}(x, y) = H(x, y, x) = x + y + 2 \log \left[ 1 + e^{-\frac{x}{2}} + e^{-x} + e^{-y} + e^{-\frac{x+y}{2}} + e^{-x-y} \right]. \quad (42)$$

There are three ways of ordering the linear forms involved in (42):

$$0 < 2y < x < x + y < x + 2y < 2x < 2x + 2y \quad (43)$$

$$0 < x < 2y < x + y < 2x < x + 2y < 2x + 2y \quad (44)$$

$$0 < x < 2x < x + y < 2y < x + 2y < 2x + 2y. \quad (45)$$

We can point out several issues with these orders. In each of these orders $(x, y) \mapsto x + y$ (which is the linear form that approximates $\tilde{H}$) is not one of the two smallest linear forms, hence we won't be able to write $\tilde{H}(s, t)$ as $t + \psi(s, t)$ when using our usual change of variables. This is a first issue, as Theorem 3.12 holds for functions $h$ of the form $h(s, t) = t + \psi(s, t)$. The second issue comes from the order (44): the change of variables made by taking the two smallest linear forms in this order does not map the induced cone $\{0 < x < 2y < x + y < 2x < x + 2y < 2x + 2y\}$ to a half plane of the form $\{0 < s < t\}$, while Theorem 3.12 only holds for cones of this specific form. Finally in the order (45), the two smallest linear forms are colinear, hence they don't form a basis of $(\mathbf{R}^2)^*$. Thus we cannot use these two smallest linear forms to perform a change of variables. Actually this issue can be solved easily by denoting $s = 2x$ and
$t = x + y$, but when applying Theorem 3.12 we would only be able to conclude that the convolution $s^i \star t^j(\ell)|_{\mathbb{H}_{\{0<s<t\}}}^{\tilde{H}}$ has a coefficient $-\frac{1}{4}$ in the exponential remainder term (instead of the usual $-\frac{1}{2}$).

Instead of having to deal with these issues we shall keep using the formula (42). Moreover $\tilde{H}$ only depends on two variables, so we will be able to handle the computations.

On the one hand we have a good control of the derivatives of $\tilde{H}$:

$$ \frac{\partial \tilde{H}}{\partial x}(x, y) = 1 + O(e^{-\frac{x}{2}}) \qquad (46) $$

$$ \frac{\partial \tilde{H}}{\partial y}(x, y) = 1 + O(e^{-\frac{y}{2}}) \qquad (47) $$

$$ \frac{\partial^2 \tilde{H}}{\partial x \partial y}(x, y) = O(e^{-\frac{x+y}{2}}). \qquad (48) $$

On the other hand, by taking $x_0 = 0$ in Lemma 3.8 we have $\varphi = \psi$ and we obtain:

$$ \mathcal{P}_x[\psi f] = \psi \mathcal{P}_x f - \mathcal{P}_x[\psi' \mathcal{P}_x f]. $$

Therefore the term (10) in Proposition 3.10 vanishes, thus with less restrictive assumptions on the function $h$ than in Theorem 3.11 (here we just need to assume that $h$ satisfies the hypothesis $(\mathbf{H}_n)$), we have:

$$ \mathcal{L}_{\ell}^{K} \left[ f_1 \star \cdots \star f_n(\ell) \Big|_{\varphi}^{h} \right] = \int_{\ell=h(\mathbf{x}_n)} f_1(x_1) \cdots f_{n-1}(x_{n-1}) F_n^K(\mathbf{x}_n) \varphi(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \\ + \mathcal{L}_{\ell}^{K} \left[ \int_{\ell=h(\mathbf{x}_n)} f_1(x_1) \cdots f_{n-1}(x_{n-1}) G_n(\mathbf{x}_n) \varphi(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \right] \\ - \sum_{m=0}^{K-1} \mathcal{L}_{\ell}^{K-1-m} \left[ \int_{\ell=h(\mathbf{x}_n)} f_1(x_1) \cdots f_{n-1}(x_{n-1}) H_n^m(\mathbf{x}_n) \varphi(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \right] \\ + \sum_{m=0}^{K-1} \mathcal{L}_{\ell}^{K-1-m} \mathcal{P}_{\ell} \left[ \int_{\ell=h(\mathbf{x}_n)} f_1(x_1) \cdots f_{n-1}(x_{n-1}) I_n^m(\mathbf{x}_n) \frac{\partial \varphi}{\partial x_n}(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \right]. $$

Let us denote $\tilde{\psi}(x, y) = \tilde{H}(x, y) - x - y$, $\tilde{\varphi} = \mathbb{1}_{\mathbb{R}_{>0}^2} e^{\tilde{\psi}}$ and $f_1(x) = x^i e^x$, $f_2(y) = y^j e^y$. With these notations we have:

$$ e^{\ell} \left( x^i \star y^j(\ell) \Big|_{\mathbb{1}_{\mathbb{R}_{>0}^2}}^{\tilde{H}} \right) = f_1 \star f_2(\ell) \Big|_{\tilde{\varphi}}^{\tilde{H}}. $$

When we apply the operator $\mathcal{L}_{\ell}^{K_2}$ on this convolution, we can therefore write $\mathcal{L}_{\ell}^{K_2}[f_1 \star f_2(\ell)|_{\tilde{\varphi}}^{\tilde{H}}]$ as a linear combination of (images under powers of $\mathcal{P}_{\ell}$ and $\mathcal{L}_{\ell}$ of) terms of the form:

$$ \Theta_{\Phi}(\ell) = \int_{\ell=\tilde{H}(x,y)} f_1(x) \Phi(x,y) \tilde{\varphi}(x,y) \frac{dxdy}{d\ell} = f_1 \star 1(\ell) \Big|_{\Phi\tilde{\varphi}}^{\tilde{H}}, $$

where $\Phi$ is either $F_2^{K_2}, G_2, H_2^m$ or $I_2^m \frac{\partial \tilde{\psi}}{\partial y}$. Hence, applying the operator $\mathcal{L}_{\ell}^{K_1}$ to $\Theta_{\Phi}$, we can write $\mathcal{L}_{\ell}^{K_1+K_2}[f_1 \star f_2(\ell)|_{\tilde{\varphi}}^{\tilde{H}}]$ as a linear combination (up to powers of $\mathcal{P}_{\ell}$ and $\mathcal{L}_{\ell}$) of terms of the form:

$$ \Xi_{\Psi}(\ell) = \int_{\ell=\tilde{H}(x,y)} \Psi(x,y) \tilde{\varphi}(x,y) \frac{dxdy}{d\ell}, $$

where $\Psi$ is either $F_1^{K_1}\Phi, G_1\Phi, H_1^{m'}\Phi, I_1^{m'} \frac{\partial \Phi}{\partial x}$ or $I_1^{m'} \Phi \frac{\partial \tilde{\psi}}{\partial x}$.
Fortunately, the estimaties (46), (47) and (48) imply that, as long as $K_1 \ge i+1$ and $K_2 \ge j+1$, there
exist a constant $c > 0$ and an integer $N \ge 0$ such that:

$$
|\Psi(x, y)| \leq c((1+x)(1+y))^N e^{\frac{x+y}{2}},
$$

and since $x + y \leq H(x, y)$, we can conclude that:

$$
|\Xi_{\Psi}(\ell)| \leq c(1+\ell)^{2N} e^{\frac{\ell}{2}} \int_{\ell=\tilde{H}(x,y)} \frac{dx dy}{d\ell}.
$$

Hence $\Xi_{\Psi} \in \mathcal{R}_w$ and $x^i \star y^j(\ell)|_{\mathbb{R}_{>0}^2}^{\tilde{H}} \in S\mathcal{F}_w^{i+j+1}$.

We then have the following partial conclusion:

**Lemma 4.8.** The conclusion of Lemma 4.5 holds if $\gamma$ is of type $(\mathcal{T}_0')$.

*Remark.* Curves of type $(\mathcal{T}_0')$ illustrate some limitations of our method. In order to deal with more general curves in further work, we shall first modify this method.

4.2.3. *Eight-shaped curves in a holed torus.* Let us finally make the computations when $g = n = 1$, i.e. when $\gamma$ is a curve of type $(\mathcal{T}_1)$ or $(\mathcal{T}_1')$.

* Let us assume that $\gamma$ is $(\mathcal{T}_1)$. In this case, similarly to the what happened with curves of type $(\mathcal{P}_2)$ and $(\mathcal{P}_2')$, we have for every test function $F : \mathbf{R} \to \mathbf{C}$:

$$
\mathbb{E}[F^{\gamma}] = \int_{0}^{\infty} F(h(x, x, L_1)) x \frac{dx}{V_{1,1}(L_1)},
$$

hence if we denote:

$$
\tilde{h} : x \mapsto h(x, x, L_1) = 2\operatorname{argcosh}\left(2 \cosh^2\left(\frac{x}{2}\right) + \cosh\left(\frac{L_1}{2}\right)\right),
$$

and $\ell_0(L_1) = \tilde{h}(0) = h(0, 0, L_1)$, we have:

$$
\mathbb{E}[F^{\gamma}] = \int_{\ell_0(L_1)}^{\infty} F(\ell) \tilde{h}^{-1}(\ell) \frac{d\tilde{h}^{-1}(\ell)}{d\ell} \frac{d\ell}{V_{1,1}(L_1)}.
$$

Thus we have:

$$
\forall \ell \ge 0, V_{\mathcal{O}_{\gamma}}(\ell, L) = \tilde{h}^{-1}(\ell) \frac{d\tilde{h}^{-1}(\ell)}{d\ell} \mathbf{1}_{(\ell_0(L_1), +\infty)}(\ell).
$$

By a direct computation we have:

$$
V_{\mathcal{O}_{\gamma}}(\ell, L_1) = \frac{\sinh\left(\frac{\ell}{2}\right)\operatorname{argcosh}\left(\cosh\left(\frac{\ell}{2}\right) - \cosh\left(\frac{L_1}{2}\right) - 1\right)}{2\sqrt{\left(\cosh\left(\frac{\ell}{2}\right) - \cosh\left(\frac{L_1}{2}\right) - 1\right)^2 - 1}} \mathbf{1}_{(\ell_0(L_1), +\infty)}(\ell).
$$

* If $\gamma$ is of type $(\mathcal{T}_1')$, similarly to what we computed in the previous case, we have:

$$
V_{\mathcal{O}_{\gamma}}(\ell, L_1) = \frac{2 \sinh\left(\frac{\ell}{2}\right) \operatorname{argcosh}\left(\frac{\cosh\left(\frac{\ell}{2}\right)}{1+2 \cosh\left(\frac{L_1}{2}\right)}\right)}{\sqrt{\cosh^2\left(\frac{\ell}{2}\right) - \left(1+2 \cosh\left(\frac{L_1}{2}\right)\right)^2}} \mathbf{1}_{(\ell_1(L_1), +\infty)}(\ell),
$$

where $\ell_1(L_1) = h(L_1, 0, 0)$.
5. CONCLUSION

By putting together all the partial conclusions we have been able to make until now, we just proved our main result:

**Theorem 5.1.** Let $\gamma$ be an eight-shaped curve on $S_{g,n}$. There exist functions $\{V_{\mathcal{O}_\gamma}(\cdot, L)\}_{L \in \mathbb{R}_{\ge 0}^n} \in \mathcal{S}\mathcal{F}_w^{6g-7+2n}$ supported in $[2\operatorname{argcosh}(3), +\infty)$ such that for every test function $F: \mathbf{R} \to \mathbf{C}$ we have:

$$ \mathbb{E}[F^{\gamma}] = \int_{2\operatorname{argcosh}(3)}^{\infty} F(\ell) V_{\mathcal{O}_{\gamma}}(\ell, L) \frac{d\ell}{V_{g,n}(L)}. $$

Moreover:

(i) if $\gamma$ is of type $(\mathcal{P}_0), (\mathcal{P}_1), (\mathcal{P}'_1), (\mathcal{T}_0)$ or $(\mathcal{T}'_0)$, then $V_{\mathcal{O}_\gamma}$ is a continuous function of its first variable;

(ii) if $\gamma$ is of type $(\mathcal{P}_2), (\mathcal{P}'_2), (\mathcal{T}_1)$ or $(\mathcal{T}'_1)$, $V_{\mathcal{O}_\gamma}$ is discontinuous at $h(L_i, L_j, 0), h(L_i, 0, L_j), h(0, 0, L_1)$ or $h(L_1, 0, 0)$ respectively, for some $1 \le i, j \le n, i \ne j$.

As a direct consequence:

**Corollary 5.2.** Under the same assumptions we have $a \mapsto \mathbb{E}[N_\gamma(a)] \in \mathcal{S}\mathcal{F}^{6g-6+2n}$.

*Proof.* Let us denote by $P_\gamma^L$ the polynomial part of $V_{\mathcal{O}_\gamma}(\cdot, L)$ and $r_\gamma^L: \ell \mapsto V_{\mathcal{O}_\gamma}(\ell, L) - P_\gamma^L(\ell)$. Estimate (6) implies that $r_\gamma^L \in L^1(\mathbf{R}_{\ge 0})$ and:

$$ \int_a^\infty |r_\gamma^L(\ell)| d\ell = \mathcal{O}\left((1+a)^c e^{-\frac{a}{2}}\right). $$

Hence, denoting by $Q_\gamma^L$ the primitive of $P_\gamma^L$ such that $Q_\gamma^L(0) = \int_0^\infty r_\gamma^L(\ell) d\ell$, we have:

$$ \mathbb{E}[N_\gamma(a)] = Q_\gamma^L(a) + \int_a^\infty r_\gamma^L(\ell) d\ell = Q_\gamma^L(a) + \mathcal{O}\left((1+a)^c e^{-\frac{a}{2}}\right). $$

□

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