<figure><img src="image_30.png" /><figcaption>Figure 27: Corona formed by a collection of squares.</figcaption></figure>

these norms for the special case of a connection that is piecewise constant in $\theta$ and piecewise affine in $r$, as is the case for $A_N$. Let us first recall that for $\tilde{A} \in \Omega^1(\Sigma, \mathfrak{g})$,

$$W(\tilde{A})(r, \theta) = \int_{0}^{\theta} A(0 \xrightarrow{r} \theta) d\theta.$$

and

$$\|\tilde{A}\|_{W_{r,\theta}^{\alpha,\alpha-1,p,s}} := \left( \sum_{a \in \text{Crit}(f)} \sum_{I_a \times J_a} \sum_{n=0}^{\infty} \left( 2^{n(s-\frac{2}{p})} \|S_{(r_0,\theta_0)}^{2^{-n}*} W(\tilde{A})\|_{W_{r,\theta}^{\alpha,\alpha;p}(I \times J)} \right)^p \right)^{\frac{1}{p}}, \quad (10.1)$$

where we take a sum over all critical points, and for each critical point $a$, the sum over a finite cover of the form $Ia^r \times J_a^\theta$ of a certain corona of the form $\{m \in \Sigma; \text{dist}(m, \Psi(a)) \in [1, 2]\}$ centered near a singular point $\Psi(a) = (r_0, \theta_0)$. The factor $S^{-n}$ means that we are testing closer and closer to the critical point. The Figure 27 explains the situation: The center of the grids represent the critical points, and the blue and red lattices show the base corona formed of 12 squares that we scale closer and closer to the critical point.

It is therefore enough to study $\|A_N\|_{W_{r,\theta}^{\alpha,\alpha-1,p,s}}$ near an arbitrary fixed saddle point, that we will assume has coordinates $(0, 0)$ and for a fixed square that we will call $I \times J$. Recall that $W(A_N)$ is piecewise affine in both directions, on a grid of mesh $2^{-N}$. Therefore, to study its norm on a square of mesh $2^{-n}$, we should separate two cases:

$$\begin{aligned} & \sum_{n=0}^{\infty} \left( 2^{n(s-\frac{2}{p})} \|S_{(0,0)}^{2^{-n}*} W\|_{W_{r,\theta}^{\alpha,\alpha;p}(I \times J)} \right)^p \\ & = \sum_{n \le N} \left( 2^{n(s-\frac{2}{p})} \|S_{(0,0)}^{2^{-n}*} W\|_{W_{r,\theta}^{\alpha,\alpha;p}(I \times J)} \right)^p \\ & \quad + \sum_{n > N} \left( 2^{n(s-\frac{2}{p})} \|S_{(0,0)}^{2^{-n}*} W\|_{W_{r,\theta}^{\alpha,\alpha;p}(I \times J)} \right)^p, \end{aligned}$$
<figure><img src="image_31.png" /><figcaption>Figure 28: Resolution $N$ (black) bigger than resolution $n$ (red).</figcaption></figure>

**The case $n < N$.** In this case, the problem reduces to bound the $W_{r,\theta}^{\alpha,\alpha;p}(I \times J)$ norm of a piecewise linear function in both direction. In fact, since the $N$ resolution is larger than the $n$ resolution, the situation is as in Figure 28. We have that

$$ \|S_{(0,0)}^{2^{-n}*} W\|_{W_{r,\theta}^{\alpha,\alpha;p}(I \times J)} = \int_{I_1 \times I_1} \int_{I_1 \times I_2} \frac{|A(2^{-n}\theta \xrightarrow{2^{-n}r'} 2^{-n}\theta') - A(2^{-n}\theta \xrightarrow{2^{-n}r'} 2^{-n}\theta')|^p}{|r-r'|^{1+p\alpha} |\theta-\theta'|^{1+p\alpha}} dr dr' d\theta d\theta', $$

and by a change of variables, we get

$$ \|S_{(0,0)}^{2^{-n}*} W\|_{W_{r,\theta}^{\alpha,\alpha;p}(I \times J)} = 2^{2n(1-p\alpha)} \int_{2^{-n}I_1 \times 2^{-n}I_1} \int_{2^{-n}I_2 \times 2^{-n}I_2} \frac{|A(\theta \xrightarrow{r'} \theta') - A(\theta \xrightarrow{r'} \theta')|^p}{|r-r'|^{1+p\alpha} |\theta-\theta'|^{1+p\alpha}} dr dr' d\theta d\theta'. $$

Therefore, the problem reduces to understand a 1-dimensional analogue, in which we understand the behavior of a classical one dimensional Gagliardo norms for a piecewise affine function. This is done in the next lemma.

**Lemma 10.1.** Let $M \in \mathbb{N}$. Let $f : [0,1] \to \mathfrak{g}$ be piecewise affine function such that $f$ is linear between $f(\frac{k}{M})$ and $f(\frac{k+1}{M})$ for all $k = 0, \dots, M-1$. Then,

$$ \|f\|_{s,p}^{p} := \int_{[0,1]^2} \frac{|f(t) - f(s)|_{\mathfrak{g}}^{p}}{|t-s|^{1+ps}} dt ds \le \frac{12^p}{M^{1-ps}} \sum_{0 \le k < l \le M} \frac{|f(\frac{k}{M}) - f(\frac{l}{M})|^p}{(k-l)^{1+ps}}. $$

*Proof.* We have

$$ \begin{aligned} \|f\|_{s,p}^{p} &= \sum_{0 \le k, l \le 2^N - 1} \int_{[\frac{k}{M}, \frac{k+1}{M}] \times [\frac{l}{M}, \frac{l+1}{M}]} \frac{|f(x) - f(y)|^p}{|x - y|^{1+ps}} dx dy \\ &= \sum_{0 \le k \le M-1} \int_{[\frac{k}{M}, \frac{k+1}{M}]^2} \frac{|f(x) - f(y)|^p}{|x - y|^{1+ps}} dx dy + \sum_{0 \le p \ne q \le M-1} \int_{[\frac{k}{M}, \frac{k+1}{M}] \times [\frac{l}{M}, \frac{l+1}{M}]} \frac{|f(x) - f(y)|^p}{|x - y|^{1+ps}} dx dy. \end{aligned} $$

For $x \in [\frac{k}{M}, \frac{k+1}{M}]$, and $y \in [\frac{l}{M}, \frac{l+1}{M}]$,

$$ f(x) = M(a_{k+1} - a_k) \left(x - \frac{k}{M}\right) + a_k \text{ and } f(y) = M(a_{l+1} - a_l) \left(y - \frac{l}{M}\right) + a_l. $$

Now, consider the three different cases below :
• If $k = l$, then $|f(x) - f(y)| = M|a_{k+1} - a_k||x - y|$, and

$$ \frac{|f(x) - f(y)|^p}{|x - y|^{1+ps}} \leq M^p |a_{k+1} - a_k|^p |x - y|^{p-1-ps} \leq M^{1+ps} |a_{k+1} - a_k|^p, $$

and therefore

$$ \sum_{0 \le k \le M-1} \int_{[\frac{k}{M}, \frac{k+1}{M}]^p} \frac{|f(x) - f(y)|^p}{|x - y|^{1+ps}} dxdy \le M^{ps-1} \sum_{0 \le k \le M-1} |a_{k+1} - a_k|^2 \le K. $$

• If not then

$$ |f(x) - f(y)| \le |f(x) - a_{k+1}| + |a_{k+1} - a_l| + |a_l - f(y)| \le |a_{k+1} - a_k| + |a_{k+1} - a_l| + |a_{l+1} - a_l| $$

and

$$ |f(x) - f(y)|^p \le 3^p (|a_{k+1} - a_k|^2 + |a_{l+1} - a_l|^2 + |a_{k+1} - a_l|^2) $$

since $|x - y| > (l - (k + 1))$, then

$$ \frac{|f(x) - f(y)|^p}{|x - y|^{1+ps}} \le 3^p \left( M^{1+ps} |a_{p+1} - a_p|^p + M^{1+ps} |a_{q+1} - a_q|^p + M^{1+ps} \frac{|a_{k+1} - a_l|^p}{(l - (k + 1))^{1+ps}} \right). $$

Thus,

$$ \sum_{0 \le p \ne q \le 2^{N-1}} \int_{[\frac{p}{2^N}, \frac{p+1}{2^N}] \times [\frac{q}{2^N}, \frac{q+1}{2^N}]} \frac{|f(x) - f(y)|^2}{|x - y|^{1+2s}} dxdy \le 3 \times 3^p K, $$

and finally $\|f\|_{s,p} \le 4 \times 3^p K$.

**Lemma 10.2.** For $N \ge n$, we have

$$ \int_{2^{-n}I_1 \times 2^{-n}I_1} \int_{2^{-n}I_2 \times 2^{-n}I_2} \frac{|A(\theta \xrightarrow{r'} \theta') - A(\theta \xrightarrow{r'} \theta')|^p}{|r - r'|^{1+p\alpha} |\theta - \theta'|^{1+p\alpha}} dr dr' d\theta d\theta' \\ \le \frac{12^{2p}}{2^{2N(1-ps)}} \sum_{\substack{2^{N-n} \le k \le l \le 2^{N-n+1} \\ 2^{N-n} \le p \le q \le 2^{N-n+1}}} \frac{|A_N(k2^{-N} \xrightarrow{q2^{-N}} l2^{-N}) - A_N(k2^{-N} \xrightarrow{p2^{-N}} l2^{-N})|^p}{(l-k)^{1+p\alpha} (q-p)^{1+p\alpha}}. $$

*Proof.* We apply Lemma 10.1 two times.

**The case** $n \ge N$. In this case, the resolution $N$ is bigger than the resolution $n$, as depicted in Figure 29. This case reduces to computing a classical 1-dimensional Gagliardo norm for (not anymore piecewise) affine function. In this case, we have the following proposition.

**Proposition 10.3.** We have

$$ \int_{2^{-n}I_1 \times 2^{-n}I_1} \int_{2^{-n}I_2 \times 2^{-n}I_2} \frac{|A(\theta \xrightarrow{r'} \theta') - A(\theta \xrightarrow{r'} \theta')|^p}{|r - r'|^{1+p\alpha} |\theta - \theta'|^{1+p\alpha}} dr dr' d\theta d\theta' \\ \lesssim 2^{2p(N-n)} |A(0 \xrightarrow{2^{-N}} 2^{-N}) - A(0 \xrightarrow{0} 2^{-N})|^p. $$

The proof consist in applying two times in a row the next lemma.
<figure><img src="image_32.png" /><figcaption>Figure 29: Resolution N (black) bigger than resolution n (red).</figcaption></figure>

**Lemma 10.4.** Let $f: [0, 1] \to \mathbb{R}$ be a piecewise affine function, linear on each interval $[k2^{-N}, (k+1)2^{-N}]$. Then, for $n \ge N$,

$$ \|S^{2-n} f\|_{W^{\alpha;p}} \lesssim 2^{p(N-n)} |f(2^{-N}) - f(0)|^p $$

*Proof.* We have

$$ \begin{aligned} \|S^{2-n} f\|_{W^{\alpha;p}([\frac{1}{2},1])} &= \int_{[\frac{1}{2},1]\times[\frac{1}{2},1]} \frac{|f_N(2^{-n}t) - f_N(2^{-n}s)|^p}{|t-s|^{1+p\alpha}} dsdt \\ &= 2^{p(N-n)} |f(2^{-N}) - f(0)|^p \int_{[\frac{1}{2},1]\times[\frac{1}{2},1]} |t-s|^{p-1-\alpha p} dt ds. \end{aligned} $$

<figure><img src="image_33.png" /></figure>

Finally, we obtain the desired bound in the following proposition.

**Proposition 10.5.** For all $N \ge 0$, we have

$$ \begin{aligned} \|A_N\|_{W^{\alpha;\alpha;p}(\mathrm{Cyl},\mathbb{g})} & \le \sum_{0 \le n \le N} \frac{2^{np(s-2\alpha)}}{2^{2N}(1-p\alpha)} \sum_{\substack{2^{-n+N-2} \le n < m \le 2^{-n+N-1} \\ 2^{-n+N-2} \le k, l \le 2^{-n+N-1}}} \frac{|A_N(k2^{-N} \xrightarrow{m2^{-N}} l2^{-N}) - A_N(k2^{-N} \xrightarrow{n2^{-N}} l2^{-N})|^p}{|l-k|^{1+p\alpha}(m-n)^{1+p\alpha}} \\ & \quad + 2^{2pN(s-2\alpha)} |A(0 \xrightarrow{2^{-N}} 2^{-N}) - A(0 \xrightarrow{0} 2^{-N})|^p. \end{aligned} $$

*Proof.* Combination of Lemmas 10.2 and 10.3.

<figure><img src="image_34.png" /></figure>

## 10.2 Probabilistic bounds

Based on Proposition 10.5, to be able to bound $E[\|A_N\|_{W^{\alpha;\alpha;p}(\mathrm{Cyl},\mathbb{g})}]$, we need to understand how to bound

$$ E \left[ \left| A_N(k2^{-N} \xrightarrow{m2^{-N}} l2^{-N}) - A_N(k2^{-N} \xrightarrow{n2^{-N}} l2^{-N}) \right|^p \right]. $$

Recall from the definition of $A_N$ that

$$ \begin{aligned} & A_N(k2^{-N} \xrightarrow{m2^{-N}} l2^{-N}) - A_N(k2^{-N} \xrightarrow{n2^{-N}} l2^{-N}) \\ & = \sum_{i=k}^{l} \log \left( M_{i2^{-N} \xrightarrow{m2^{-N}} (i+1)2^{-N}} \right) - \log \left( M_{i2^{-N} \xrightarrow{n2^{-N}} (i+1)2^{-N}} \right), \end{aligned} $$
where $M$ is sampled under $\mu^N$. Therefore, we should understand, for $n < m \in \mathbb{N}$, $l \in \mathbb{N}$, and independent random variables $X_1, \dots, X_n, Y_1, \dots, Y_{m-n}$ such that $X_i \sim \mu_{s_i}$, and $Y_i \sim \mu_{t_i}$ the quantity

$$ \mathbb{E}\left[\left|\sum_{j=1}^{l} \log(X_1 \cdots X_n Y_1 \cdots Y_{m-n}) - \log(X_1 \cdots X_n)\right|^p\right]. $$

In fact, we have the following estimate.

**Proposition 10.6.** Consider a family of probability measures $(\mu_t)_{t>0}$ verifying $M_\beta$ for some $\beta > 0$. Let moreover $\varepsilon > 0$.

There exists a constant $C_\beta$ such that for integers $n < m$ and $l$, and independent random variables $(X_u^j)_{\substack{1 \le j \le l \\ 1 \le u \le n}}$ and $(Y_v^j)_{\substack{1 \le j \le l \\ 1 \le v \le m-n}}$ such that

$$ \mathbb{P}(X_u^j \in dg) = \mu_{s_u^j}(dg) \text{ and } \mathbb{P}(Y_v^j \in dg) = \mu_{t_v^j}(dg), $$

where

$$ \max \left( \max_{\substack{1 \le j \le l \\ 1 \le u \le n}} s_u^j, \max_{\substack{1 \le j \le l \\ 1 \le v \le m-n}} t_u^j \right) \le \varepsilon, $$

we have

$$ \mathbb{E}\left[\left|\sum_{j=1}^{l} \log(X_1^j \cdots X_n^j Y_1^j \cdots Y_{m-n}^j) - \log(X_1^j \cdots X_n^j)\right|^{2\beta}\right] \le C_\beta l^\beta \left((m-n)^\beta \varepsilon^\beta + n^{2\beta-\varepsilon^2\beta}\right). $$

We need several lemmas for the proof.

**Lemma 10.7.** Under the $M_\beta$ condition, there exists a constant $C > 0$ such that for all $0 \le s_1, \dots, s_n, t_1, \dots, t_m \le 1$, we have

$$ \int_{G^{n+m}} |\log(x_1 \cdots x_n y_1 \cdots y_m) - \log(y_1 \cdots y_m)|^{2\beta} \prod_{i=1}^{n} \mu_{s_i}(dx_i) \prod_{j=1}^{m} \mu_{t_j}(dy_j) \\ \le C_p((s_1 + \cdots + s_n)^\beta + (t_1 + \cdots + t_m)^{2\beta}). $$

*Proof.* By the Baker–Campbell–Hausdorff formula, there exists a constant $C > 0$ such that for all $x, y \in G$,

$$ |\log(xy) - \log(x) - \log(y)| \le C(|\log x|^2 + |\log y|^2). $$

This gives

$$ |\log(xy) - \log(x)|^{2\beta} \le 3^{2\beta} C^{2\beta} (|\log y|^{2\beta} + |\log x|^{4\beta} + |\log y|^{4\beta}). $$

By taking the integral, we get

$$ \begin{align*} & \int_{G^{n+m}} |\log(x_1 \cdots x_n y_1 \cdots y_m) - \log(y_1 \cdots y_m)|^{2\beta} \prod_{i=1}^{n} \mu_{s_i}(dx_i) \prod_{j=1}^{m} \mu_{t_j}(dy_j) \\ & \lesssim \int_{G^{n+m}} |\log(x_1 \cdots x_n)|^{2\beta} \prod_{i=1}^{n} \mu_{s_i}(dx_i) \prod_{j=1}^{m} \mu_{t_j}(dy_j) \\ & + \int_{G^{n+m}} |\log(x_1 \cdots x_n)|^{4\beta} \prod_{i=1}^{n} \mu_{s_i}(dx_i) \prod_{j=1}^{m} \mu_{t_j}(dy_j) \\ & + \int_{G^{n+m}} |\log(y_1 \cdots y_m)|^{4\beta} \prod_{i=1}^{n} \mu_{s_i}(dx_i) \prod_{j=1}^{m} \mu_{t_j}(dy_j) \\ & \lesssim (s_1 + \cdots + s_n)^\beta + (t_1 + \cdots + t_m)^{2\beta}, \end{align*} $$

where the $2\beta^-$ comes from Lemma 8.2. $\square$

<figure><img src="image_35.png" /></figure>
**Lemma 10.8.** Using the same notations as Proposition 10.6, let us define, for $1 \le k \le l$,

$$Z_k := \sum_{j=1}^{k} \log(X_1^j \cdots X_n^j Y_1^j \cdots Y_{m-n}^j) - \log(X_1^j \cdots X_n^j).$$

The sequence $(Z_k)_{1 \le k \le l}$ is a martingale.

*Proof.* For fixed $k$, $Z_k$ is the sum of independent random variables. We only need to check that $Z_k$ is centered. Since the convolution of measures that are invariant by conjugation and inversion is again invariant by conjugation and inversion, we only need to show that $\int_{G^2} (\log(ab) - \log(a))\mu(da)\nu(db) = 0$. for two measures $\mu$ and $\nu$ invariant by conjugation and inversion. However, we have

$$ \begin{aligned} & \int_{G^2} (\log(ab) - \log(a))\mu da \nu(db) = \int_{G^2} (\log(ab^{-1}) - \log(a))\mu da \nu(db) \\ &= \int_{G^2} (\log((ba^{-1})^{-1}) - \log(a))\mu da \nu(db) = - \int_{G^2} (\log(ba^{-1}) - \log(a^{-1}))\mu da \nu(db) \\ &= - \int_{G^2} (\log(ba) - \log(a))\mu da \nu(db) = - \int_{G^2} (\log(aba^{-1}a) - \log(a))\mu da \nu(db) \\ &= - \int_{G^2} (\log(ab) - \log(a))\mu da \nu(db). \end{aligned} $$

Therefore $Z_k$ is a martingale as the sum of centered independent random variables. $\square$

Recall below one form of the BDG inequality.

**Lemma 10.9 (BDG inequality for vector valued martingales).** Let E be an euclidean space and $(Z_k)_{k \ge 0}$ be a $E$-valued martingales. Then, for all $p > 1$, there exists a constant $c_p$ such that

$$\forall n \in \mathbb{N}, \mathbb{E} \left[ \sup_{k \le n} \|Z_k\|^p \right] \le c_p \mathbb{E} \left[ \left( \sum_{k \le n} \|Z_k - Z_{k-1}\|^2 \right)^{\frac{p}{2}} \right].$$

*proof of Proposition 10.6.* By Lemma 10.9, we get

$$ \begin{aligned} \mathbb{E}[|Z_l|^{2\beta}] & \le C_{\beta} \mathbb{E} \left[ \left( \sum_{i=1}^{l-1} |Z_{i+1} - Z_i|^2 \right)^{\beta} \right] \le C_{\beta} l^{\beta-1} \sum_{i=1}^{l-1} \mathbb{E} \left[ |Z_{i+1} - Z_i|^{2\beta} \right] \\ & \le C_{\beta} l^{\beta} \sup_{1 \le j \le k} \mathbb{E} \left[ \left| \log(X_1^j \cdots X_n^j Y_1^j \cdots Y_{m-n}^j) - \log(X_1^j \cdots X_n^j) \right|^{2\beta} \right] \\ & \le C_{\beta} l^{\beta} \left( (m-n)^{\beta} \varepsilon^{\beta} + n^{2\beta-\varepsilon 2\beta-1} \right), \end{aligned} $$

where the last step comes from Lemma 10.7. $\square$

We finally show the following main proposition of this subsection.

**Proposition 10.10.** We have, for all integers $k \le n \le m \le l$,

$$ \begin{aligned} \mathbb{E} \left[ \left| A_N(k2^{-N} \xrightarrow{m2^{-N}} l2^{-N}) - A_N(k2^{-N} \xrightarrow{n2^{-N}} l2^{-N}) \right|^p \right] & \le \sup_{\substack{k \le j \le l \\ n \le i \le m}} \sigma(\Box(i2^{-N}, (i+1)2^{-N}, j2^{-N}, (j+1)2^{-N}))^\beta (l-k)^\beta (m-n)^\beta \\ & \quad + \sup_{\substack{k \le j \le l \\ n \le i \le m}} \sigma(\Box(i2^{-N}, (i+1)2^{-N}, j2^{-N}, (j+1)2^{-N}))^{2\beta} (l-k)^\beta n^{2\beta}. \end{aligned} $$

*Proof.* It is a direct application of Proposition 10.6. $\square$
## 10.3 Tightness result

In this section, we will show the tightness result.

**Proposition 10.11.** To show that $(A_N)_{N \ge 0}$ is tight, it is enough to show

$$ \sup_{N \ge 0} \mathbb{E}[\|A_N\|_{W^{\alpha;\alpha;p}(\text{Cyl},\mathfrak{g})}] < \infty. $$

*Proof.* We show in the Proposition C.4 from the appendix that for $\alpha > \alpha'$, we have the compact embedding of $W^{\alpha;\alpha;p}(\text{Cyl}, \mathfrak{g})$ in $W^{\alpha';\alpha;p}(\text{Cyl}, \mathfrak{g})$. Therefore, the ball $B_{W^{\alpha;\alpha;p}(\text{Cyl}, \mathfrak{g})}(0, c)$, seen as a subset in $W^{\alpha';\alpha;p}(\text{Cyl}, \mathfrak{g})$ is compact, and is of measure arbitrarily close to 1, provided we choose $c$ big enough. $\square$

**Proposition 10.12.** For $s < \frac{1}{2}$, We have

$$ \sup_{N \ge 0} 2^{pN(s-2\alpha)} \mathbb{E} \left| A\left(0 \xrightarrow{2^{-N}} 2^{-N}\right) - A\left(0 \xrightarrow{0} 2^{-N}\right) \right|^p < \infty $$

*Proof.* We have from Proposition 10.10,

$$ \mathbb{E} \left| A\left(0 \xrightarrow{2^{-N}} 2^{-N}\right) - A\left(0 \xrightarrow{0} 2^{-N}\right) \right|^p \lesssim \sigma(\Box(0, 2^{-N}, 0, 2^{-N}))^{\frac{p}{2}} \le 2^{-\frac{Np}{2}} $$

which concludes the result.

**Proposition 10.13.** We have

$$ \sup_{2^{N-n-2} \le k, l \le 2^{N-n-1}} \sigma(\Box(k2^{-N}, (k+1)2^{-N}, l2^{-N}, (l+1)2^{-N})) \lesssim 2^{-2N+n}. $$

*Proof.* It is a direct consequence of Lemma 6.1.

**Lemma 10.14.** Let $u < 0$, and $\alpha \in (0, 1)$. Then, for any integers $A < B$, we have

* if $u > -1$, there exists a constant $C(u) > 0$ such that $\sum_{A \le p < q \le B} (q-p)^u \le C(u) (B-A)^{u+2}$,

* if $u \le -1$, there exists a constant $c(u) > 0$ such that $\sum_{A \le p < q \le B} (q-p)^u \ge c(u) (B-A)$.

* there exists $C(u, \alpha) > 0$ such that

$$ \sum_{A \le p < q \le B} \frac{p^u}{(q-p)^{1+\alpha u}} \le C(u, \alpha) (B-A)^{1+u-\alpha u}. $$

**Theorem 10.15.** For $\alpha < \frac{1}{2}$ and $s < \frac{1}{2}$, the sequence $(A_N)_{N \ge 0}$ is tight in the weighted space $W^{\alpha;\alpha;p}(\text{Cyl}, \mathfrak{g})$.

*Proof.* Using the previous results of the current subsection, we only still need to show that the first sum of *prop say* is bounded in $N$. We have

$$ \begin{aligned} & \sum_{0 \le n \le N} \frac{2^{np(s-2\alpha)}}{2^{2N}(1-p\alpha)} \sum_{\substack{2^{-n+N-2} \le n < m \le 2^{-n+N-1} \\ 2^{-n+N-2} \le k, l \le 2^{-n+N-1}}} \frac{\mathbb{E} \left| A_N(k2^{-N} \xrightarrow{m2^{-N}} l2^{-N}) - A_N(k2^{-N} \xrightarrow{n2^{-N}} l2^{-N}) \right|^p}{|l-k|^{1+p\alpha} (m-n)^{1+p\alpha}} \\ & \le \sum_{0 \le n \le N} \frac{2^{np(s-2\alpha)}}{2^{2N}(1-p\alpha)} \sum_{\substack{2^{-n+N-2} \le n < m \le 2^{-n+N-1} \\ 2^{-n+N-2} \le k, l \le 2^{-n+N-1}}} 2^{\frac{p}{2}(n-2N)} (k-l)^{\frac{p}{2}-1-p\alpha} (m-n)^{\frac{p}{2}-1-p\alpha} \\ & \quad + \sum_{0 \le n \le N} \frac{2^{np(s-2\alpha)}}{2^{2N}(1-p\alpha)} \sum_{\substack{2^{-n+N-2} \le n < m \le 2^{-n+N-1} \\ 2^{-n+N-2} \le k, l \le 2^{-n+N-1}}} 2^{p(n-2N)} (k-l)^{\frac{p}{2}-1-p\alpha} \frac{n^p}{(m-n)^{1+p\alpha}}. \end{aligned} $$
<figure><img src="image_36.png" /><figcaption>Figure 30: Measure on the closed surface.</figcaption></figure>

Using the previous lemma, we have that for $\alpha < \frac{1}{2}$, the first term is bounded by

$$ \sum_{n \ge 0} 2^{n(ps - \frac{p}{2} - 2)}, $$

which is finite for $s < \frac{1}{2}$ and the second goes to 0 as $N \to \infty$. □

# 11 Proof of the main theorem

## 11.1 From the cylinder back to the surface

*proof of Proposition 1.2.* We already managed the singular part of A formed of unstable currents. We only need to explain what happens in the bulk part. We identified the limit in Section 9, and we showed the tightness in $W^{\alpha;\alpha;p}(\text{Cyl}; \mathfrak{g})$ in Proposition 10.11. Since $W^{\alpha;\alpha;p}(\text{Cyl}; \mathfrak{g})$ is continuously embedded in $C^{\alpha - \frac{1}{p}; \alpha - \frac{1}{p}; p}(\text{Cyl}; \mathfrak{g})$, we have

$$ A_N \frac{C^{\alpha - \frac{1}{p}; \alpha - \frac{1}{p}; p}(\text{Cyl}; \mathfrak{g})}{N \to \infty} \to \partial_{\theta} \langle \xi, 1_{\square(r, \theta)} \rangle \, d\theta. $$

Therefore, taking $p$ large enough, we get from Proposition 5.7,

$$ \Psi^* A_N \frac{C^{\frac{1}{2} - (S)'}{N \to \infty} \to \Psi^* \left( \partial_{\theta} \langle \xi, 1_{\square(r, \theta)} \rangle \right) d\theta. $$

Note that the space $C^{\frac{1}{2} - (S)'}$ is just a global space. However, locally, functional spaces look exactly like on the cylinder, and we have better local convergence properties. □

## 11.2 Closing the Surface

In the previous section, we worked with the free boundary Yang-Mills measure. This means we considered a surface with one outgoing boundary, with no prescribed boundary conditions. In the current section, we would like to see how we can adapt this construction to the case of a closed surface. The main idea is to condition the holonomy on the boundary to be equal to $1_G$.

Consider a closed surface $\Sigma$, and a Morse function on $\Sigma$ verifying the Morse-Smale condition. Blow-up the surface on the maximum to get a new surface, that we will call $S$, with boundary $\partial S$. Note that $\partial S$ is now exactly the maximum set of the morse function. The idea is that the Yang-Mills measure on the closed surface is the Yang-Mills measure on the blown-up surface, in which we condition the holonomy on $\partial\Sigma$ to be $1_G$; this is summarized in Figure 30. In this section, we will study this conditioning.

We will start in the lattice, where conditioning is much easier handled, and then show that the sequence of conditioned measure converges. We will more precisely prove the following theorem.
Let $(\mu_t)_t$ be a family of probability measures verifying the conditions (H). Recall the definition of the sequence of random connections $(A^{(N)}_{(M,U)})_{N \ge 0}$ introduced in Section 10. Let $\mathcal{B}$ be a Banach space of distributions in which $(A^{(N)}_{(M,U)})_{N \ge 0}$ converges.

**Theorem 11.1.** For each $N \ge 1$, let $(M_N(\cdot|\cdot|))_{[g] \in G \setminus \mathrm{Ad}(G)}$ be the disintegration of the law of $A^{(N)}_{(M,U)}$ with respect to $\mathrm{Hol}(A^{(N)}_{(M,U)}, \partial\Sigma)$, seen as a random variable valued in $G \setminus \mathrm{Ad}(G)$. Then, in $\mathcal{M}(\mathcal{B})$,

$$ \forall g \in G, M_N(\cdot|\cdot|) \xrightarrow[N \to \infty]{} \mathbb{M}(\cdot|\cdot|). $$

### 11.2.1 Disintegrating the lattice measure

Consider again the Morse lattice on $\Sigma$ of resolution $2^{-N}$. Consider a Morse-Gauge fixed Yang-Mills measure associated to a family $\mu$ of probability measure. This means that we are dealing with two kinds of random variables:

* the variables $M_{\theta \xrightarrow{r} \theta_+}$ for $\theta \in \Theta_-^N$ and $r \in \mathbb{R}^N$,

* the $2g$ independent uniform random variables $(U_b)_{b=1}^{2g}$ in $G$.

The idea is to disintegrate the lattice measure with respect to the holonomy $\mathrm{Hol}(\max f)$. The problem is that applying the standard disintegration theorem yields a family of conditional measures defined only *almost everywhere*, which is insufficient for our purposes. To overcome this, we establish a Bayes-type formula: the law of any observable depending on the bonds below $f = \max f - \epsilon$, conditioned on $\mathrm{Hol}(\max f) = 1_G$, equals the average over $g$ of the conditional expectation of the observable given $\mathrm{Hol}(\max f - \epsilon) = g$, multiplied by a transition kernel of the form $\mathbb{P}(\mathrm{Hol}(\max f) = 1_G | \mathrm{Hol}(\max f - \epsilon) = g)$. Since we now only require an average over $g$, the almost-everywhere disintegration of the observable along the level sets $\mathrm{Hol}(\max f - \epsilon) = g$ becomes sufficient. In the sequel we will give a way of formalizing this idea.

Let us fix a $j \in \Theta_-^N$, v with $j \ge 2g + 1$, and introduce

$$ H_{N;j} := \mathrm{Hol}(0 \xrightarrow{j} 2\pi). $$

For $r \in \mathbb{R}^N$, let us define the sigma algebras

$$ \mathcal{F}_{\le r}^N := \sigma \left( \left\{ M_{\theta \xrightarrow{\rho} \theta_+}; \theta \in \Theta_{-}^N, \rho \le r \right\} \cup \{U_b; b \le 2g\} \right), \quad \mathcal{F}_{\ge r}^N := \sigma \left( \left\{ M_{\theta \xrightarrow{\rho} \theta_+}; \theta \in \Theta_{-}^N, \rho \ge r \right\} \cup \{U_b; b \ge r\} \right), $$

and

$$ \mathcal{F}_r^N := \sigma \left( \left\{ M_{\theta \xrightarrow{r} \theta_+}; \theta \in \Theta_-^N \right\} \cup \{U_b; b \le 2g\} \right). $$

The idea is that the sigma algebra $\mathcal{F}_{\le r}^N$ (resp. $\mathcal{F}_{\ge r}^N$, resp $\mathcal{F}_r^N$) contain the information on all the bonds of the lattice $\Lambda^N$ situated below the level $f = r$ (rep. above $f = r$, resp. exactly at $f = r$), in addition to all bonds included in the zero area bands. This is why $r$ is always supposed to be above the last critical point.

First, we would like to understand what is the law of the holonomy of the circle $0 \xrightarrow{k2^{-N}} 2\pi$ at diacid level $k2^{-N}$ knowing all the bonds situated below level $j2^{-N}$. This is the purpose of the following lemma.

**Lemma 11.2.** Let $f: G \to \mathbb{R}$ be a measurable central function, and let $j \le k$. Then

$$ \mathbb{E}[f(H_{N;k})|\mathcal{F}_{\le j}] = \int_G f(g)p_N(H_{N;j}^{-1}g)dg, \quad \text{where } p_N(x) = \sum_{\lambda \in \hat{G}} \left( \prod_{\substack{j \le r \le k \\ \theta \in \Theta_-^N}} \frac{\hat{\mu}_{\sigma(\square(r,r_+,\theta,\theta_+))}}{d_\lambda} \right) \chi_\lambda(x). $$
<figure><img src="image_37.png" /><figcaption>Figure 31: A part of the lattice which represents $U_1$, $M_{I_1}$, and $M_{I'_1}$.</figcaption></figure>

Proof. Let $j \le k$, we have $H_{N;k} = \prod_{b=0}^{2g-1} U_b M_{I_b} U_b^{-1} M_{I'_b}$, where each $M_{I_b}$ is a product of

$$M_{\theta \xrightarrow{k} \theta_+} \left( M_{\theta \xrightarrow{j} \theta_+} \right)^{-1} M_{\theta \xrightarrow{j} \theta_+}.$$

The Figure 31 represents $U_1$, $M_{I_1}$, and $M_{I'_1}$. Therefore, $H_{N;k}$ is a word $\mathbf{w}$ of some variables that are $F_{\le j}$ measurable, and others that are independent of $\mathcal{F}_{\le j}$. Therefore,

$$\mathbb{E}[f(H_{N;k})|\mathcal{F}_j] = \Lambda \left( \left( M_{\theta \xrightarrow{r} \theta_+} \right)_{r \le j, \theta \in \Theta_N^N}, (U_b)_{0 \le b \le 2g-1} \right),$$

where

$$\Lambda \left( \left( m_{\theta \xrightarrow{r} \theta_+} \right)_{r \le j, \theta \in \Theta_N^N}, (u_b)_{0 \le b \le 2g-1} \right) = \mathbb{E}[f(\mathbf{w}')],$$

where $\mathbf{w}'$ is the same word as $\mathbf{w}$ where we have replaced the $\mathcal{F}_{\le j}$-measurable variables by deterministic constants represented by small letter. By ad-invariance of the $M_{\theta \xrightarrow{k} \theta_+} \left( M_{\theta \xrightarrow{j} \theta_+} \right)^{-1}$, we have

$$\Lambda \left( \left( M_{\theta \xrightarrow{r} \theta_+} \right)_{r \le j, \theta \in \Theta_N^N}, (U_b)_{0 \le b \le 2g-1} \right) = \mathbb{E} \left[ f \left( h_j \prod_{\theta \in \Theta_N^N} M_{\theta \xrightarrow{k} \theta_+} \left( M_{\theta \xrightarrow{j} \theta_+} \right)^{-1} \right) \right] = \int_G f(g) p_N(h_j^{-1} g) dg,$$

where $p_N$ is the density of the convolution of all the $M_{\theta \xrightarrow{k} \theta_+} \left( M_{\theta \xrightarrow{j} \theta_+} \right)^{-1}$, and therefore given in Fourier representation by

$$p_N(x) = \sum_{\lambda \in \hat{G}} \left( \prod_{\substack{j \le r \le k \\ \theta \in \Theta_N^N}} \frac{\hat{\mu}_{\sigma(\square(r, r_+, \theta, \theta_+))}}{d_{\lambda}} \right) \chi_{\lambda}(x).$$
□

An important observation of the previous lemma is that the law of $H_{N;k}$ knowing $\mathcal{F}_{\le j}$ is only a function of $H_{N;j}$, not of all the bonds situated under the level set $j2^{-N}$. Therefore, we conclude the following lemma.

**Lemma 11.3.** We have, for any central function, $\mathbb{E}[f(H_{N;k})|\mathcal{F}_j] = \mathbb{E}[f(H_{N;k})|H_{N;j}]$.

*Proof.* Direct consequence of the previous lemma.

□

Now, we want to compute the conditional law of $H_{N;j}$ with respect to $H_{N;k}$. A first step is to understand the joint law of these two random variables.

**Lemma 11.4.** We have, for all measurable functions $f$ and $g$

$$ \mathbb{E}[f(H_{N;j})g(H_{N;k})] = \int_{G^2} f(x)g(yx)f_{H_{N;j}}(x)p_N(y)dxdy, $$

where

$$ p_N(y) = \sum_{\lambda \in \hat{G}} \left( \prod_{\substack{j \le r \le k \\ \theta \in \Theta_{-}^N}} \frac{\hat{\mu}_{\sigma(\Box(t, t_+ m \theta, \theta_+))}}{d_{\lambda}} \right) \chi_{\lambda}(y). $$

As a result, we get the desired conditional law.

**Lemma 11.5.** We have, for all measurable functions $f$

$$ \mathbb{E}[f(H_{N;j})|H_{N;k}] = \int_G f(x)\kappa_N(H_{N;k}, x)dx, \text{ where } \kappa_n(g, x) := \frac{f_{H_{N;j}}(x)p_N(gx^{-1})}{f_{H_{N;k}}(g)}. $$

Before stating the main proposition of this subsection, let us first define precisely the support of a function in $C_b(\mathcal{B}, \mathbb{R})$.

**Definition 11.6.** Let $F \in C_b(\mathcal{B}, \mathbb{R})$. A subset $K$ of $\Sigma$ is said to be in the support of $F$ if

$$ \forall \varphi \in C^{\infty}(\Sigma), F(\varphi + h) = F(\varphi), \forall h \in C_c^{\infty}(\Sigma \setminus K). $$

The support of $F$ is the closure of the union of all sets in the support of $F$.

**Proposition 11.7.** Let $F \in C_b(\mathcal{B})$ such that the support of $F$ is included in $\{f \le j\}$. For $k > j$, we have
We have

$$ \mathbb{E}[F(A_N)|H_{N;k} = 1_G] = \int_G \mathbb{E}[F(A_N)|H_{N;j} = x] \kappa_N(1_G, x)dx. $$

*Proof.* For a central function $f$,

$$ \mathbb{E}[F(A_N)f(H_{N;k})] = \mathbb{E}[\mathbb{E}[F(A_N)f(H_{N;k})|\mathcal{F}_{\ge j}]] = \mathbb{E}[\mathbb{E}[F(A_N)|\mathcal{F}_{\ge j}]f(H_{N;k})], $$

where we have used that $H_{N;k}$ is $\mathcal{F}_{\ge j}$-measurable. Since $F$ is $\mathcal{F}_{\le j}$-measurable, then

$$ \begin{align*} \mathbb{E}[F(A_N)f(H_{N;k})] &= \mathbb{E}[\mathbb{E}[F(A_N)|\mathcal{F}_j]f(H_{N;k})] = \mathbb{E}[\mathbb{E}[\mathbb{E}[F(A_N)|\mathcal{F}_j]f(H_{N;k})|\mathcal{F}_j]] \\ &= \mathbb{E}[\mathbb{E}[F(A_N)|\mathcal{F}_j]\mathbb{E}[f(H_{N;k})|\mathcal{F}_j]] = \mathbb{E}[\mathbb{E}[F(A_N)|\mathcal{F}_j]\mathbb{E}[f(H_{N;k})|H_{N;j}]] \\ &= \mathbb{E}[\mathbb{E}[\mathbb{E}[F(A_N)|\mathcal{F}_j]|H_{N;j}]\mathbb{E}[f(H_{N;k})|H_{N;j}]] = \mathbb{E}[\mathbb{E}[F(A_N)|H_{N;j}]\mathbb{E}[f(H_{N;k})|H_{N;j}]]. \end{align*} $$

We can now write

$$ \mathbb{E}[f(H_{N;k})|H_{N;j}] = \int_G f(g)p_N(H_{N;j}^{-1}g)dg, $$
which gives

$$
\begin{align*}
\mathbb{E}[F(A_N)f(H_{N;k})] &= \mathbb{E} \left[ \mathbb{E}[F(A_N)|H_{N;j}] \int_G f(g)p_N(H_{N;j}^{-1}g)dg \right] \\
&= \int_G \int_G \mathbb{E}[F(A_N)|H_{N;j} = x] f(g)p_N(x,g)\mathbb{P}(H_{N;j} \in dx)dg \\
&= \int_G f(g)\mathbb{P}(H_{N;k} \in dg) \int_G \mathbb{E}[F(A_N)|H_{N;j} = k] \kappa_N(g,k)dk.
\end{align*}
$$

Therefore,

$$
\mathbb{E}[F(A_N)|H_{N;k} = 1_G] = \int_G \mathbb{E}[F(A_N)|H_{N;j} = x] \kappa_N(1_G, x) dx.
$$

<figure><img src="image_38.png" /></figure>

11.2.2 Convergence of the conditioned measures

A sequence of measures on $\mathcal{B} \times G$. The main intermediate results to prove the convergence of the
conditioned measures is to study the convergence of the couple $(A_{(M,U)}^{(N)}, \text{Hol}(A_{(M,U)}^{(N)}, \partial\Sigma))$.

Indeed, from the free boundary measure $\mu_N$ together with the holonomy map $\text{Hol}(A_N, \partial\Sigma)$ (which is a random matrix when $A_N$ is chosen randomly under the probability measure $\mu_N$), we define a measure $\mathbf{m}_N$ on $\mathcal{B} \times G$. This measure describes the joint law of the pair $(A_N, \text{Hol}_{\partial\mathcal{S}_{\text{out}}}(A_N)) \in \mathcal{B} \times G$.

What we must prove is the following statement :

**Lemma 11.8.** The sequence of measures **m**<sub>*N*</sub> converges to a limiting measure **m** describing the joint law of (A, Hol<sub>∂S<sub>out</sub></sub>(A)) ∈ 𝒩 × G where A is chosen randomly under the limiting measure µ.

Proof. For every $N$, there is a well-defined holonomy map $\text{Hol}_{\partial S_{\text{out}}}$ which is described as the solution of $dU_N = U_N dW_N$, where

$$
dW_N = \sum_{\theta \in \Theta_{-N}} \log \left( M_{\theta \xrightarrow{\max f} \theta_+^N} \right) 2^N 1_{[\theta, \theta_+]}(\theta) d\theta,
$$

and $W_N$ is an affine interpolation of a $\mathfrak{g}$-valued random walk that converges as $N \to +\infty$ to a re-
parametrized $\mathfrak{g}$-valued Brownian motion. Since we have

$$
\forall p \in \mathbb{N}, \sup_{\substack{N \ge 0 \\ \theta \in \Theta_{-N}}} \mathbb{E} \left[ \left| 2^{N/2} \log \left( M_{\theta \xrightarrow{\max f} \theta_{+}^{N}} \right) \right|^p \right] < +\infty,
$$

we deduce that **W**<sub>N</sub> → **W**<sup>g</sup> in the rough path topology for variations of order 2 by the proof of Bayer-Friz [6, Thm 3.3 p. 269]. Indeed, this reference proves a Donsker Theorem in the rough path setting for walks which are independent but not necessarily identically distributed, itself being a generalization of [18, Thm 1 p. 3489].

Let us recall the proof. Consider the geometric rough path

$$
\mathbf{W}_N := \left( W_N(s,t) \in \mathfrak{g}; \int_s^t W_N \otimes dW_N \in \mathfrak{g} \otimes \mathfrak{g} \right).
$$

Let

$$
\mathbf{W}^{\mathfrak{g}} = \left( W_{s,t}, \int_s^t W \otimes \circ dW \right)
$$

denote the $\mathfrak{g}$-valued enhanced Brownian motion in the sense of Stratonovich. We would like to show
that the pair $(A_N, \mathbf{W}_N)$ converges in law in $\mathcal{B} \times \mathcal{C}^\alpha$ where the second space is the space of rough paths
of Hölder regularity $\alpha \in (0, \frac{1}{2})$. Let us start by the easy part. The pair $A_N, W_N := \int_{\{\max(f)\} \times [0, \theta]} A_N$ converges in law where

$$dW_N = \sum_{\theta \in \Theta_N} \log(M_{\theta \xrightarrow{\max(f)} \theta_N}) 2^N 1_{[\theta, \theta_N]}(\theta) d\theta$$

and $W_N$ is an affine interpolation of a $\mathfrak{g}$-valued random walk that converges to some re-parametrized $\mathfrak{g}$-valued Brownian motion as $N \to +\infty$. To show the convergence in law of the pair $A_N, U_N(2\pi)$ using the continuity RDE, we need to enhance $W_N$ to the rough path $\mathbf{W}_N$. The strategy is to reduce the proof of the usual central limit Theorem and prove some kind of tightness in Hölder spaces of rough paths. The first idea is to consider the interpretation of rough paths as Lie group valued paths. Start from the Lie algebra $\mathfrak{g}$, and consider any rough path as some element of the tensor algebra :

$$\mathbf{W} := (1, W_{s,t}, \mathbb{W}_{s,t}) \in \mathbb{R} \oplus \mathfrak{g} \oplus \mathfrak{g} \otimes \mathfrak{g} = T^2\mathfrak{g}.$$

This forms a non-commutative tensor algebra whose unit reads $(1, 0, 0)$ [42, section 2.3 p. 17]. The elements of the form $(1, b, c) \in T^{(2)}(\mathfrak{g})$ forms a group denoted by $T_1^{(2)}(\mathfrak{g})$ sitting inside $T^{(2)}(\mathfrak{g})$. This group is called the step-2 nilpotent Lie group. Note that $\mathbf{W}_{s,t} = \mathbf{W}_{0,t} \circ \mathbf{W}_{0,s}^{-1}$ is therefore a rough paths and can be interpreted as a path valued in the Lie group $T_1^{(2)}(\mathfrak{g})$; and $\mathbf{W}_{s,t}$ is just an increment of the above path. This group has a natural Carnot–Carathéodory distance $d_C$ and geometric rough paths of Hölder regularity $\alpha$ can be identified with Hölder maps of regularity $\alpha$ valued in $T_1^{(2)}(\mathfrak{g})$ for the distance $d_C$ [42, ]. The idea is to prove some sort of Donsker theorem and the corresponding CLT for random walks valued into the nilpotent step 2 group. If we have some centered, independent $\mathfrak{g}$-valued random variables $(\xi_i^n, 1 \le i \le n)$, we denote by $e^{\xi_i^{(n)}} := (1, \xi_i, \xi_i \otimes \xi_i)$ the corresponding element in $T_1^{(2)}(\mathfrak{g})$. Assume that, for all $v, w \in \mathfrak{g}$,

$$\mathbb{E} \left( \left\langle v, \xi_i^{(n)} \right\rangle_{\mathfrak{g}} \left\langle w, \xi_i^{(n)} \right\rangle_{\mathfrak{g}} \right) = a_i^{(n)} \langle v, w \rangle_{\mathfrak{g}} + o(\frac{1}{n}), \text{ with } a_i^{(n)} \sim \frac{1}{n}, \sum_{i=1}^{n} a_i^{(n)} = 1.$$

Then by the CLT proved in [6, Lemma 4.1 p. 271], we know that $e^{\xi_1^{(n)}} \otimes \cdots \otimes e^{\xi_n^{(n)}}$ converges in law to the time 1 of the Brownian motion on $T_1^{(2)}(\mathfrak{g})$ whose infinitesimal generator is the natural left-invariant sub-Laplacian $\sum_{i=1}^{\dim(\mathfrak{g})} X_i^2$ in $T_1^{(2)}(\mathfrak{g})$. Then the next step consists in reducing to some combinatorial estimate. One key idea is that a geodesic $t\xi$ in $\mathfrak{g}$ lifts uniquely to a geodesic $e^{t\xi}$ in $T_1^{(2)}(\mathfrak{g})$ for the Carnot–Carathéodory distance. So linear (geodesic) interpolation on $\mathfrak{g}$ lifts functorially to geodesic interpolation in $T_1^{(2)}(\mathfrak{g})$ for the Carnot–Carathéodory distance [42, 3.6 p. 40].

The hard part of the proof is the control of tightness in the Hölder space $C^\alpha$ which is essential for rough differential equations. By independence of increments and left invariance of the Carnot–Carathéodory distance, it is enough to prove that

$$\sup_n \mathbb{E} (d_C (\mathbf{W}_u, \mathbf{W}_v)^a) \le c|u-v|^{1+b} \quad (11.1)$$

for the ratio $\frac{b}{a}$ close to $\frac{1}{2}$ and $[u, v] = [0, \sum_{i=1}^k a_i^{(n)}]$. This rewrites

$$\mathbb{E} \left( \| e^{\xi_1^{(n)}} \otimes \cdots \otimes e^{\xi_k^{(n)}} \|^a \right) \le \left( \sum_{i=1}^{n} a_i^{(n)} \right)^{1+b}$$

that we need to prove. By changing the scaling again, we are reduced to proving an estimate of the form :

$$\mathbb{E} \left( \| e^{\xi_1} \otimes \cdots \otimes e^{\xi_k} \|^2 p \right) \lesssim \left( \mathbb{E} \left( \sum_{i=1}^k \| \xi_i \|^2_{\mathfrak{g}} \right) \right)^{4p}$$
for any sequence $\xi_1, \dots, \xi_n, \dots$ of $\mathfrak{g}$-valued independent centered random variables, for all $p \ge 2$ and $k$ large enough. The proof of this bound is exactly the content of [6, Prop 4.3 p. 273] and this tells us the sequence $(A_N, \mathbf{W}_N) \in \mathcal{B} \times \mathcal{C}^\alpha$ converges in law to the pair $(A, \mathbf{W}) \in \mathcal{B} \times \mathcal{C}^\alpha$ where the second factor $\mathbf{W}_N \to \mathbf{W}$ as $\mathcal{C}^\alpha$ geometric rough path for $\alpha \in (\frac{1}{3}, \frac{1}{2})$. Since $U_N$ solves the SDE $dU_N = U_N \circ dW_N$ in Stratonovich form, we can deduce by Ito-Lyons continuity of the RDE in the driving signal, that the pair $(A_N, U_N)$ converges in law to some pair $(A, U) \in \mathcal{B} \times \mathcal{G}$. $\square$

**Proof of the first part of Theorem 11.1.** Let $\epsilon > 0$, and $F \in C_b(\mathcal{B})$, such that $\text{supp } F \subset \{f \le \max f - \epsilon\}$. Choose $N$ large enough, and pick a dyadic level $j$ in $\{\max f - 2\epsilon \le f \le \max f - \epsilon\}$. We have

$$ \begin{aligned} \mathbb{E}[F(A_N)|H_{N;\max f} = 1_G] &= \frac{1}{f_{H_{N;\max f}}(1_G)} \int_G p_N(x) \mathbb{E}[F(A_N)|H_{N;j} = x] \mathbb{P}(H_{N;j} \in dx) \\ &= \frac{1}{f_{H_{N;\max f}}(1_G)} \int_G (p_N(x) - p_\infty(x)) \mathbb{E}[F(A_N)|H_{N;j} = x] \mathbb{P}(H_{N;j} \in dx) \\ &+ \frac{1}{f_{H_{N;\max f}}(1_G)} \int_G p_\infty(k) \mathbb{E}[F(A_N)|H_{N;j} = x] \mathbb{P}(H_{N;j} \in dx). \end{aligned} $$

The first term

$$ \left| \int_G (p_N(x) - p_\infty(x)) \mathbb{E}[F(A_N)|H_{N;j} = x] \mathbb{P}(H_{N;j} \in dx) \right| \le \|p_N - p_\infty\|_\infty \|F\|_\infty \to 0, $$

as a direct consequence of Theorem 7.2, and by Lemma 11.8, the second term

$$ \int_G p_\infty(x) \mathbb{E}[F(A_N)|H_{N;j} = x] \mathbb{P}(H_{N;j} \in dx) = \mathbb{E}[F(A_N)p_\infty(H_{N;j})] \to \mathbb{E}[F(A)p_\infty(H_j)], \square $$

### 11.2.3 Control on Hölder-Besov norms

We would like to control the Hölder-Besov regularities of the conditioned random connections. The first direct result is that

$$ A_{(M,U)}^{(N)}|\text{Hol}(A_{(M,U)}^{(N)}, \partial\Sigma) = [g] \text{ converges in } H^{-(1+\epsilon)}(\Sigma, \mathfrak{g}). $$

Now, we will control the convergence in $C_{r,\text{loc}}^\alpha C_\theta^{\alpha-1}$, for all $\alpha \in (0, \frac{1}{2})$. This finishes the proof of Theorem 1.1.

**Proof of Theorem 1.1.** Set $\alpha \in (0, \frac{1}{2})$, and let $\psi$ be any function in $C_c^\infty(\Sigma \setminus \partial\Sigma)$. Then we choose $m$ big enough, and $j = 2^m \max(f) - 1$ so that $F := A \in \mathcal{B} \mapsto \|A\psi\|_{C_r^\alpha C_\theta^{\alpha-1}} \in \mathbb{R}$ be $F_{\le j}^{(N)}$ measurable, for all $N \ge m$. From Theorem 10.11, we have

$$ \sup_{N \ge 0} \mathbb{E} [F(A_{(M,U)}^{(N)})] \le \sup_{N \ge 0} \mathbb{E} [\|A_{(M,U)}^{(N)}\|] < \infty, $$

where we have used the continuity of the multiplication by $\psi$ in $C_r^\alpha C_\theta^{\alpha-1}$. Therefore,

$$ \int_G \mathbb{E} [F(A_{(M,U)}^{(N)}) | H_j^{(N)} = g] \mathbb{P}(H_1^{(N)} \in dg) \le C $$

which gives, using the $C^\infty$ convergence of $\mathbb{P}(H_1^{(N)} \in dg)$ to $p_{\sigma(\Sigma)}(g)dg$ (direct consequence of Theorem 7.2), which is bounded below by a non-zero positive number,

$$ \int_G \mathbb{E} [F(A_{(M,U)}^{(N)}) | H_j^{(N)} = g] dg \le C'. $$
Now, this concludes from Proposition 11.7 that

$$ \sup_{N \ge 0} \mathbb{E} \left[ F \left( A_{(M,U)}^{(N)} \right) \mid \mathrm{Hol}(A_{(M,U)}^{(N)}, \partial \Sigma) = 1_G \right] < \infty. $$

This means that for every test function $\psi \in C_c^\infty(\Sigma \setminus \partial\Sigma)$, the sequence $A_{(M,U)}^{(N)}\psi$ is tight in $C_{r,loc}^\alpha C_\theta^{\alpha-1}$. Since the uniqueness of the limit has already been established in the previous section, we have finally that the sequence $A_{(M,U)}^{(N)}\psi$ converges in $C_{r,loc}^\alpha C_\theta^{\alpha-1}$. $\square$

# A Classical results in harmonic analysis

## A.1 Path-wise integral of some rough differential forms

In this appendix, we record a lemma that we occasionally use, which allows for the path-wise integration of rough differential forms. We also take this opportunity to note that the integration of differential forms is currently an active area of research, with many recent and interesting developments. We mention, in particular, the works of Züst [75], Alberti-Stepanov-Trevisan[2] ; and very recently Chandra-Singh [9], and Jaffard [48].

**Lemma A.1.** Let $M$ be some $C^\infty$ manifold. Let $T \in \mathcal{D}'(M)$ be a top degree current whose coefficients are in the Hölder-Besov space $C^\alpha$ for $\alpha > -1$. Then for every domain $\Omega \subset M$ which has smooth boundary, the pairing

$$ \int_{\Omega} T = \int_{M} T 1_{\Omega} $$

is well-defined. It is bilinear continuous when we endow the indicator function $1_\Omega$ with the topology of $B_{1,1}^{-\alpha}$.

*Proof.* Without loss of generality using partitions of unity, the invariance of Hölder-Besov spaces by diffeomorphisms and adapted charts, we reduce to some half-space $\{(x_1, \dots, x_n); x_1 \ge 0\}$ in $\mathbb{R}^n$ and we want to give meaning to $\int_{\mathbb{R}^n} T\chi\Theta(x_1)$ where $\chi$ is the Heaviside function. The delta distribution $\delta_{\{0\}}$ in 1d belongs to the Besov space $B_{1,\infty}^0$ hence in $B_{1,1}^{-\varepsilon}$ for all $\varepsilon > 0$. Therefore by taking primitive, the Heaviside function should belong to $B_{1,1,loc}^{1-\varepsilon}$ for all $\varepsilon > 0$. It follows that the product $\chi\Theta(x_1)$ belongs to the Besov space $B_{1,1}^{1-\varepsilon}$ for all $\varepsilon > 0$ since $\chi \in C_c^\infty$ is a multiplier for Besov spaces. Then we conclude using $C^\alpha = B_{\infty,\infty}^\alpha$ and by the duality in Besov spaces $(B_{1,1}^{-\alpha})' = B_{\infty,\infty}^\alpha$. Beware that the duality is only in the sense stated, we do not have the converse : $(B_{\infty,\infty}^\alpha)' \neq B_{1,1}^{-\alpha}$. $\square$

## A.2 Restriction in Besov spaces

**Lemma A.2 (Restriction Lemma in Besov regularity).** Under the assumption of Lemma A.1, the product $T1_\Omega$ belongs to the Besov space $B_{1,1}^\alpha$.

Note that there is a serious loss in regularity since the space $B_{1,1}^\alpha$ is much larger than the Hölder space $B_{\infty,\infty}^\alpha$.

*Proof.* We use $1_\Omega \in B_{1,1}^{1-\varepsilon}$. We use the product decomposition of Bony $T1_\Omega = T \succ 1_\Omega + T \prec 1_\Omega + T \circ 1_\Omega$ where $T \succ 1_\Omega \in B_{1,1}^\alpha$, $T \prec 1_\Omega \in B_{1,1}^{1-\varepsilon}$, $T \circ 1_\Omega \in B_{1,1}^{\alpha+1-\varepsilon}$ by the usual rules on the paraproduct and resonant products acting on Besov spaces as proved in [58, Lemma 2.1.34 p. 40] from the thesis of Martin which generalizes certain statements from [11]. This allows to conclude. $\square$
### A.3 Young products of distributions and duality

Since we are often using it in our work, we need to recall the Young criterion for multiplying Hölder-Besov distributions with Hölder functions.

**Lemma A.3 (Young product of Hölder functions and distributions).** The product of smooth functions

$$ (f_1, f_2) \in C^\infty(\mathbb{R}^n) \times C^\infty(\mathbb{R}^n) \mapsto f_1 f_2 \in C^\infty(\mathbb{R}^n) $$

extends uniquely as a bilinear continuous map from $C^\alpha(\mathbb{R}^n) \times C^\beta(\mathbb{R}^n) \mapsto C^{\inf(\alpha,\beta)}(\mathbb{R}^n)$ for $\alpha + \beta > 0$.

A consequence of the above is that for all compact set $K \subset \mathbb{R}^n$, the set $C_K^\alpha(\mathbb{R}^n)$ of Hölder distributions of regularity $\alpha < 0$ and supported by $K$ injects continuously in the dual of $C^\beta(\mathbb{R}^n)$ provided $\alpha + \beta > 0$.

*Proof.* The proof can be found in [44] and follows immediately from [44, Lemma 2.1 p. 11], see also [11, Thm 2.52] and [42, thm 13.16 p. 201]. □

## B Recollection on the spectral analysis of Morse–Smale flows

Let us collect the needed tools from [34]. Our spectral analysis of Morse–Smale flows is very much inspired by the microlocal approach developed by Faure and Sjöstrand to study the correlation spectrum of Anosov flows [39], which contain geodesic flows on negatively curved manifolds. We briefly describe the general strategy. We will denote by $(x; \xi)$ elements of the cotangent $T^*\Sigma$ where $x$ (resp) denotes position (resp momentum). For a given function $m(x; \xi)$ in $S^0(T^*\Sigma)$, sometimes also called symbol of order 0, we define the following Sobolev space of distributions of variable order

$$ \mathcal{H}^m := \mathbf{Op} \left( (1 + |\xi|_{g(x)}^2)^{m(x;\xi)} \right) L^2(\Sigma) $$

where **Op** means a quantization of symbols on the cotangent of space $T^*\Sigma$ of the surface $\Sigma$, it is chosen in such a way that $\mathbf{Op}((1 + |\xi|_{g(x)}^2)^{m(x;\xi)})$ is a formally self-adjoint operator with principal symbol $(1 + |\xi|_{g(x)}^2)^{m(x;\xi)}$.

Denote by $H_V := \langle \xi, V(x) \rangle \in C^\infty(T^*\Sigma)$ the Hamiltonian derived from the vector field $V = \nabla f$ and by $(\Phi_{H_V}^t)_{t \in \mathbb{R}}: T^*\Sigma \mapsto T^*\Sigma$ the corresponding Hamiltonian flow. The order function $m$ is constructed in such a way that away from the zero section, it has good decay property along the forward Hamiltonian flow, there exists $R > 0$ s.t. for all $|\xi|_{g(x)} \ge R$:

$$ X_{H_V} \log(1 + |\xi|_{g(x)}^2) m(x; \xi) \le -c < 0. $$

For our purpose, we shall need to extend the above definition on distributions to currents. This extension relies on tools from Hodge theory. We consider the bundle $\Lambda^k T^*\Sigma \mapsto \Sigma$ of $k$ forms. Given some order function $m$ as above, we define $(1 + |\xi|_{g(x)}^2)^{m(x;\xi)} \mathbf{Id} \in C^\infty(T^*\Sigma, \text{End}(\Lambda^k T^*\Sigma))$ which is the product of some symbol on $T^*\Sigma$ with the identity section $\mathbf{Id} \in C^\infty(\text{End}(\Lambda^k T^*\Sigma))$ acting on the bundle of $k$-forms. Using the Hodge star $\star$, we can define a scalar product on $C^\infty(\Lambda^k T^*\Sigma)$ as $\langle \alpha, \beta \rangle := \int_{\Sigma} \alpha \wedge \star\beta$, the completion for this scalar product defined the space $L^2(\Lambda^k T^*\Sigma)$, or one could use the induced metric on the bundle $\Lambda^k T^*\Sigma$ of $k$ forms itself. Then the Sobolev space of currents of order $m$ and degree $k$ is defined as

$$ \mathcal{H}^{m;k}(\Sigma) := \mathbf{Op} \left( (1 + |\xi|_{g(x)}^2)^{-m(x;\xi)} \mathbf{Id} \right) L^2 \left( \Lambda^k T^* \Sigma \right) \quad (B.1) $$

where $\mathbf{Op}((1 + |\xi|_{g(x)}^2)^{-m(x;\xi)} \mathbf{Id})$ is formally self-adjoint with principal symbol $(1 + |\xi|_{g(x)}^2)^{-m(x;\xi)} \mathbf{Id}$, we refer to [36, Appendix C.1] for a recollection on pseudodifferential operators acting on bundles.

One can state the main results of [34] as follows:
**Theorem B.1.** Assume $\Sigma$ is a smooth closed compact surface, $f$ a Morse function on $\Sigma$ with adapted metric $g$ such that $V = \nabla f$ satisfies the Smale transversality condition and $1$ is the only Lyapunov exponent of $V = \nabla f$. For any $L > 0$, there exists an order function $m \in C^\infty(T^*\Sigma)$ in the previous sense such that for all $k \in \{0, 1, 2\}$, the resolvent

$$ (\mathcal{L}_V + z)^{-1} : \mathcal{H}^{m,k}(\Sigma) \mapsto \mathcal{H}^{m,k}(\Sigma) $$

has a meromorphic extension to the half-space $\Re(z) > -L$. Moreover for $\Re(z) > 0$, the resolvent $(\mathcal{L}_V + z)^{-1}$ acting on $C^\infty(\Lambda^k T^*\Sigma)$ is given by the formula

$$ (\mathcal{L}_V + z)^{-1} \psi := \int_0^\infty e^{-tz} (\varphi_f^{-t*} \psi) dt \qquad (B.2) $$

for all $\psi \in C^\infty(\Lambda^k T^*\Sigma)$.

For every pair $\psi_1 \in \mathcal{H}^{-m,k}$, $\psi_2 \in \mathcal{H}^{m,2-k}$, we have the following asymptotic expansion :

$$ \int_{\Sigma} \psi_1 \wedge \varphi_f^{-t*} \psi_2 = \sum_{a \in \mathrm{Crit}(f)} \left( \int_{\Sigma} \psi_1 \wedge U_a \right) \left( \int_{\Sigma} S_a \wedge \psi_2 \right) + O \left( e^{-t} \|\psi_1\|_{\mathcal{H}^{-m,k}} \|\psi_2\|_{\mathcal{H}^{m,2-k}} \right) \quad (B.3) $$

where $(U_a)_{a \in \mathrm{Crit}(f)}$ (resp $(S_a)_{a \in \mathrm{Crit}(f)}$) are anisotropic Sobolev currents supported on the unstable manifolds $\overline{W}^u(a)$, $a \in \mathrm{Crit}(f)$ (resp stable manifolds $\overline{W}^s(a)$, $a \in \mathrm{Crit}(f)$).

Let us also give a proof of a convergence result that we used in the construction of polar coordinates.

**Lemma B.2.** Let $\gamma$ be a smooth, oriented, closed curve in $\Sigma \setminus \min(f)$ transverse to $V$. We denote by $[\gamma]$ the corresponding current of integration induced in the blow-up surface $S$. Then

$$ \varphi_f^{t*}[\gamma] \to \sum_{a \in \mathrm{Crit}(f)_1} \left( \int_{\Sigma} U_a \wedge [\gamma] \right) S_a + \left( \int_{\Sigma} [\gamma_0] \wedge [\gamma] \right) [\partial S_{in}] $$

when $t \to +\infty$ where the convergence on the r.h.s. holds in the sense of currents in the blow up surface $S$.

We greatly acknowledge Nguyen Bac Dang for explaining to us how to use cohomological arguments to prove convergence of dynamical correlators.

*Proof.* In fact we already know from the work [34] that $\varphi_f^{t*}[\gamma] \to \sum_{a \in \mathrm{Crit}(f)} (\int_{\Sigma} U_a \wedge [\gamma]) S_a \in \mathcal{D}'(\mathrm{int}(S))$ where the convergence holds in the sense of currents in $\mathcal{D}'(\mathrm{int}(S))$, these are dual to test forms which are supported in the interior of the surface. In fact in the interior, we know more: convergence takes place in suitable anisotropic Sobolev spaces of currents.

What we need to control is what happens at the boundary $\partial S$. We consider homology of currents on $S$ which is relative w.r.t. the ingoing boundary $\partial S_{in}$ and absolute w.r.t. the outgoing boundary $\partial S_{out}$. Concretely, it means that for us a current $T \in \mathcal{D}'(\mathcal{S})$ is closed if $\mathrm{supp}(\partial T) \subset \partial S_{in}$.

For this we use a topological argument, first $[\gamma]$ defines a closed current of degree 1, so we can decompose the corresponding cohomology class in the basis spanned by $([S_a], [\partial S_{in}])_{a \in \mathrm{Crit}(f)_1}$. A dual basis for the Lefschetz pairing is $(U_a, [\gamma_0])_{a \in \mathrm{Crit}(f)_1}$. By a cohomological argument based on duality, we use the fact that de Rham cohomology is finite dimensional and Lefschetz pairing is non degenerate, yields that the closed current $\sum_{a \in \mathrm{Crit}(f)_1} (\int_{\Sigma} U_a \wedge [\gamma]) S_a + (\int_{\Sigma} [\gamma_0] \wedge [\gamma]) [\partial S_{in}]$ belongs to the same cohomology class as $[\gamma]$.

From this, we deduce that $[\gamma] - (\sum_{a \in \mathrm{Crit}(f)_1} (\int_{\Sigma} U_a \wedge [\gamma]) S_a + (\int_{\Sigma} [\gamma_0] \wedge [\gamma]) [\partial S_{in}]) = dT$ is an exact current. Now we make the following elementary observation, if we remove $\gamma \cup \cup_{a \in \mathrm{Crit}(f)_1} W^s(a)$ from $S$, we see that $dT = 0$ in $\mathcal{D}'(\mathrm{int}(S) \setminus (\gamma \cup \cup_{a \in \mathrm{Crit}(f)_1} W^s(a)))$. By the constancy Theorem, this means that $T$
is locally constant on each connected component of int($\mathcal{S}$) \ ($\gamma \cup \cup_{a \in \text{Crit}(f)_1} W^s(a)$) with bounded value. Since $[\gamma] - (\sum_{a \in \text{Crit}(f)_1} (\int_{\Sigma} U_a \wedge [\gamma]) S_a + (\int_{\Sigma} [\gamma_0] \wedge [\gamma]) [\partial \mathcal{S}_{in}])$ is an integral current of finite mass and $T$ is defined uniquely up to constant, we know that up to subtracting a constant we may assume that $T = 0$ in the connected component of $\partial \mathcal{S}_{out}$.

Now $T$ is a bounded integer valued function which is vanishing near $\partial \mathcal{S}_{out}$ hence $\varphi^{t*}T \to 0$ in $D'(\mathcal{S})$ when $t \to +\infty$. There are many ways to prove such result. Observe that $\varphi^{t*}T, t \ge 0$ forms a bounded family of $L^2$ functions, hence by compactness of $L^2$ for the weak* topology, we may assume that we have a convergent subsequence. Denote by $T_\infty \in L^2$ any weak limit, its support should be invariant by the backward flow and does not contain a neighborhood of $\partial \mathcal{S}_{out}$, so $\text{supp}(T_\infty)$ should be contained in $\cup_{a \in \text{Crit}(f)_1} W^s(a) \cup \partial \mathcal{S}_{in}$ which has measure 0, $T_\infty \in L^2$ is a measurable function hence $T_\infty = 0$.

We can conclude by using the commutation between pull-back and $d$ that :

$$ \varphi_f^{t*[\gamma]} = \varphi_f^{t*[dT]} + \varphi_f^{t*[\sum_{a \in \text{Crit}(f)_1} (\int_{\Sigma} U_a \wedge [\gamma]) S_a + (\int_{\Sigma} [\gamma_0] \wedge [\gamma]) [\partial \mathcal{S}_{in}])]} \\ \xrightarrow{t \to +\infty} \sum_{a \in \text{Crit}(f)_1} (\int_{\Sigma} U_a \wedge [\gamma]) S_a + (\int_{\Sigma} [\gamma_0] \wedge [\gamma]) [\partial \mathcal{S}_{in}] $$

since $\varphi_f^{t*[dT]} = d\varphi_f^{t*[T]} \to 0$ when $t \to +\infty$ and the current

$$ \sum_{a \in \text{Crit}(f)_1} (\int_{\Sigma} U_a \wedge [\gamma]) S_a + (\int_{\Sigma} [\gamma_0] \wedge [\gamma]) [\partial \mathcal{S}_{in}] $$

is invariant.

<figure><img src="image_39.png" /></figure>

## C Continuous and compact injections in anisotropic spaces

In this appendix, we prove the necessary results on continuous and compact injections in our anisotropic spaces of currents.

We start by the continuous injections.

**Proposition C.1.** Let $\alpha \in (0, \frac{1}{2})$. If $\frac{1}{p} < \alpha$, we have the following continuous injection

$$ W_{r,\theta}^{\alpha,\alpha-1,p}(\text{Cyl}, \mathfrak{g}) \subset \mathcal{C}_{r,\theta}^{\alpha-\frac{1}{p},\alpha-1-\frac{1}{p}}(\text{Cyl}, \mathfrak{g}) $$

when we restrict to functions which are 0 at $r = 0$.

*Proof.* The proof is a simple consequence of the classical continuous injections in 1 dimension. Without loss of generality, we may prove the continuous injection in the case

$$ W_{r,\theta}^{\alpha,\alpha,p}(\text{Cyl}, \mathfrak{g}) \subset \mathcal{C}_{r,\theta}^{\alpha-\frac{1}{p},\alpha-\frac{1}{p}}(\text{Cyl}, \mathfrak{g}). $$

Then taking derivatives in the sense of distributions in the $\theta$ variable recovers the result we want to establish. The second observation is that if we restrict to distributions that vanish at $r = 0$, we can drop the $L^p$ norm in the definition of the norm and only work with the Sobolev semi-norm. For any smooth function $f$ on some interval $I = [a, b]$ which vanishes at $a$, we always have the a priori estimates, for all $\alpha \in (0, 1)$,

$$ \|f\|_{W^{s,p}(I)} = \left( \int_{I^2} \frac{|f(s_1) - f(s_2)|^p}{|s_1 - s_2|^{1+\alpha p}} ds_1 ds_2 \right)^{\frac{1}{p}} \ge C_I \sup_{s_1 \ne s_2} \frac{|f(s_1) - f(s_2)|}{|s_1 - s_2|^{\alpha - \frac{1}{p}}} $$
where the constant $C_I$ does not depend on $f$. This estimate always holds true by the injection of homogeneous Sobolev spaces into their homogeneous Hölder counterparts [51, Thm 7.23 p. 275]. Recall we need to introduce some cut-off function $\psi \in C_c^\infty(I_2)$ in the $\theta$ variable which allows to localize on some interval $I_2$.

Let us apply this estimate twice,

$$ \begin{aligned} \|f\|_{W_{r,\theta}^{\alpha,\alpha;p}(I_1 \times I_2)}^p &= \int_{I_1^2} \frac{\|f\psi(r_1, \cdot) - f\psi(r_2, \cdot)\|_{W_{\theta}^{\alpha,p}(I_2)}^p}{|r_1 - r_2|^{1+\alpha p}} dr_1 dr_2 \\ &\ge C_{I_2} \int_{I_1^2} \sup_{\theta_1 \ne \theta_2} \frac{|f\psi(r_1, \theta_1) - f\psi(r_2, \theta_1) - f\psi(r_1, \theta_2) + f\psi(r_2, \theta_2)|^p}{|r_1 - r_2|^{1+\alpha p} |\theta_1 - \theta_2|^{\alpha p-1}} dr_1 dr_2 \\ &\ge \frac{C_{I_2}}{|\theta_1 - \theta_2|^{\alpha p-1}} \int_{I_1^2} \frac{|f\psi(r_1, \theta_1) - f\psi(r_1, \theta_2) - f\psi(r_2, \theta_1) + f\psi(r_2, \theta_2)|^p}{|r_1 - r_2|^{1+\alpha p}} dr_1 dr_2 \end{aligned} $$

for all $\theta_1 \ne \theta_2$. Therefore,

$$ \begin{aligned} \|f\|_{W_{r,\theta}^{\alpha,\alpha;p}(I_1 \times I_2)}^p &\ge \frac{C_{I_2}}{|\theta_1 - \theta_2|^{\alpha p-1}} \|f\psi(\cdot, \theta_1) - f\psi(\cdot, \theta_2)\|_{W_{r,I_1}^{s,p}}^p \\ &\ge \sup_{r_1 \ne r_2} \frac{C_{I_1} C_{I_2} |f\psi(r_1, \theta_1) - f\psi(r_1, \theta_2) - (f(r_2, \theta_1) - f(r_2, \theta_2))|^p}{|\theta_1 - \theta_2|^{\alpha p-1} |r_1 - r_2|^{\alpha p-1}} \\ &\ge \frac{C_{I_1} C_{I_2} |f\psi(r_1, \theta_1) - f\psi(r_1, \theta_2) - (f\psi(r_2, \theta_1) - f\psi(r_2, \theta_2))|^p}{|\theta_1 - \theta_2|^{\alpha p-1} |r_1 - r_2|^{\alpha p-1}} \end{aligned} $$

for all $r_1 \ne r_2$, $\theta_1 \ne \theta_2$. Then taking again the sup over $\theta$ then the sup over $r$ yields the final estimates

$$ \|f\|_{W_{r,\theta}^{\alpha,\alpha;p}(I_1 \times I_2)}^p \ge C_{I_1} C_{I_2} \|f\|_{C_{r,\theta}^{\alpha,\alpha;p}(I_1 \times I_2)}^p. $$

Once we glue together the estimates thanks to the cut-off functions in $\theta$, this concludes the proof of continuous injections. $\square$

We prove another continuous injection which compares the anisotropic spaces $C^{\alpha, \alpha-1}$ with the classical non anisotropic Hölder-Besov spaces.

**Lemma C.2.** For every $\alpha \in (0,1)$, any element $T \in C_{x,y}^{\alpha,\alpha-1}(I \times J)$ for every $\chi \in C_c^\infty(J)$, the product $T1_\chi$ which is well-defined also belongs to the Hölder space $C^{\alpha-1}(I \times J)$.

*Proof.* We will use the following characterization of the Hölder-Besov space of negative regularity by scalings [42, p. 201 especially remark 13.15]. $T\chi \in C^{\alpha-1}$ if and only if for all compact balls $B$,

$$ \sup_{|\varphi|_{C_c^1(B)}=1} \sup_{a \in B} \sup_{n \ge 1} |2^{n(\alpha-1)} \langle S_a^{2-n*} (T\chi), \varphi \rangle| \le C_B. $$

The fact that $T\chi$ belongs to the anisotropic space already tells us that for all $\varphi \in C_c^1(J)$:

$$ \sup_{x \in I} \sup_{y_0 \in J} |\langle T\chi(x, 2^{-n}(\cdot - y_0), \varphi) | \le C 2^{-n(\alpha-1)} \|\varphi\|_{C^1}. $$

Therefore for $a = (x_0, y_0)$,

$$ \begin{aligned} |\langle S_a^{2-n*} (T\chi), \varphi \rangle| &= |\langle (T\chi)(2^{-n}(\cdot - x_0), 2^{-n}(\cdot - y_0)), \varphi \rangle| \\ &= |\int_{x \in I} \langle (T\chi)(2^{-n}(x - x_0), 2^{-n}(\cdot - y_0)), \varphi(x, \cdot) \rangle_y dx| \\ &\le \int_{x \in I} |\langle (T\chi)(2^{-n}(x - x_0), 2^{-n}(\cdot - y_0)), \varphi(x, \cdot) \rangle_y dx| \\ &\le C_B |I| 2^{n(1-\alpha)} \sup_{x \in I} \|\varphi(x, \cdot)\|_{C^1(J)} \end{aligned} $$
where the constant on the r.h.s. does not depend on $a \in B$ which concludes the proof of the Lemma.

For the second claim, we use the fact that $T$ is a continuous function of $x$ valued in distributions of $y$ of regularity $\alpha - 1$ hence Lemma A.1 allows us to make sense of $T1_J$ which is still continuous of $x$ valued in distributions of $y$ of regularity $\alpha - 1$. $\square$

We next prove compact injections of the anisotropic Sobolev and Hölder spaces.

**Proposition C.3.** For $\varepsilon > 0$, the injection $W^{\alpha+\varepsilon, \alpha-1+\varepsilon, p}(I \times \mathbb{S}^1) \hookrightarrow W^{\alpha, \alpha-1, p}(I \times \mathbb{S}^1)$ is compact

*Proof.* Recall that given a bounded sequence $(\varphi_n)_n$ in $L^p(\mathbb{S}^1)$ such that $\text{supp}(\widehat{\varphi}_n) \subset B(0, R)$ then this sequence has convergent subsequence in $L^p(\mathbb{S}^1)$, this is trivial since we deal with finite dimensional vector spaces.

First principle, when we multiply $T$ defined on $I \times J$ with a cut-off function $\chi \in C_c^\infty(J)$ where $\chi$ depends only on the second variable, the function $T\chi$ can be considered as living on $I \times \mathbb{S}^1$.

$$ \|T\chi\|_{W^{\alpha, \alpha, p}} := \int_{I^2} \frac{\|T\chi(t_1, \cdot) - T\chi(t_2, \cdot)\|_{W_{\theta}^{\alpha, p}}^p}{|t_2 - t_1|^{1+\alpha p}} dt_1 dt_2 $$

where

$$ \|T\chi(t_1, \cdot) - T\chi(t_2, \cdot)\|_{W_{\theta}^{\alpha, p}}^p = \sum_{j=0}^{\infty} 2^{jp\alpha} \|\psi_j(\sqrt{-\Delta})T\chi(t_1, \cdot) - T\chi(t_2, \cdot)\|_{L^p}^p. $$

Assume that we have a bounded sequence $(T_n)_n$ in the anisotropic space $W^{\alpha+\varepsilon, \alpha+\varepsilon, p}$, then note that $t \mapsto \widehat{T_n\chi}(t, k)$ is bounded sequence of functions in $W^{\alpha+\varepsilon, p}(I)$ by the a priori bound

$$ \int_{I^2} \frac{|\widehat{T\chi}(t_1, k) - \widehat{T\chi}(t_2, k)|^p}{|t_2 - t_1|^{1+\alpha p}} dt_1 dt_2 \lesssim \|T\chi\|_{W^{\alpha, \alpha, p}} \langle k \rangle^{-\alpha-\varepsilon} $$

since

$$ |\langle T_n\chi(t_1, \cdot) - T_n\chi(t_2, \cdot), e^{ik} \rangle| \le C \|T_n\chi(t_1, \cdot) - T_n\chi(t_2, \cdot)\|_{W_{\theta}^{\alpha+\varepsilon, p}} \langle k \rangle^{-\alpha-\varepsilon} $$

where $p^{-1} + q^{-1} = 1$.

Assume $\alpha - \frac{1}{p} > 0$. By a diagonal extraction argument using the compact injection $W^{\alpha+\varepsilon, p}(I) \hookrightarrow W^{\alpha, p}(I)$ where $I$ a compact interval [69, item 4) Thm 2.1]<fnref n="5" />, we can extract a subsequence so that for every $k$, $\widehat{T_n\chi}(., k)$ converges in $W^{\alpha+\varepsilon, p}$ to a limit $T_\infty(., k)$. We need to prove that this is a Cauchy sequence in $W^{\alpha, \alpha, p}$.

By boundedness of the sequence $(T_n\chi)_n$ in $W^{\alpha+\varepsilon, \alpha+\varepsilon, p}$

$$ \sup_n \int_{I^2} \frac{\sum_{j=0}^{\infty} 2^{jp(\alpha+\varepsilon)} \|\psi_j(\sqrt{-\Delta})(T_n\chi(t_1, \cdot) - T_n\chi(t_2, \cdot))\|_{L^p}^p}{|t_2 - t_1|^{1+(\alpha+\varepsilon)p}} dt_1 dt_2 \le C, $$

we deduce that for every integer $N \ge 1$, $\delta > 0$, for all $n$:

$$ \int_{I^2} \frac{\sum_{j=N}^{\infty} 2^{jp\alpha} \|\psi_j(\sqrt{-\Delta})(T_n\chi(t_1, \cdot) - T_n\chi(t_2, \cdot))\|_{L^p}^p}{|t_2 - t_1|^{1+\alpha p}} dt_1 dt_2 \le C 2^{-Np\varepsilon}, $$

and

$$ \int_{|t_2 - t_1| \ge \delta} \frac{\sum_{j=0}^{\infty} 2^{jp\alpha} \|\psi_j(\sqrt{-\Delta})(T_n\chi(t_1, \cdot) - T_n\chi(t_2, \cdot))\|_{L^p}^p}{|t_2 - t_1|^{1+\alpha p}} dt_1 dt_2 \le C \delta^{\varepsilon p}. $$

<footnote n="5">⁵In fact, it is a bit indirect to get the form we need. So first given a function $u \in W^{s,p}(I)$ extend it as $\tilde{u} \in W^{s,p}(\mathbb{R})$, the extension is linear continuous by [62, Thm 5.4 p. 548] and then apply [69, item 4) Thm 2.1] to get the compact injection</footnote>
Now using the above two bounds we can bound the following difference term

$$
\begin{align*}
& \|(T_{n_1} - T_{n_2})\chi\|_{W^{\alpha, \alpha, p}} \\
&= \int_{I^2} \frac{\sum_{j=0}^{\infty} 2^{jp\alpha} \|\psi_j(\sqrt{-\Delta})(T_{n_1}\chi(t_1, \cdot) - T_{n_1}\chi(t_2, \cdot) - T_{n_2}\chi(t_1, \cdot) + T_{n_2}\chi(t_2, \cdot))\|_{L^p}^p}{|t_2 - t_1|^{1+\alpha p}} dt_1 dt_2 \\
&\le \int_{I^2} \frac{\sum_{j=0}^{N} 2^{jp\alpha} \|\psi_j(\sqrt{-\Delta})(T_{n_1}\chi(t_1, \cdot) - T_{n_1}\chi(t_2, \cdot) - T_{n_2}\chi(t_1, \cdot) + T_{n_2}\chi(t_2, \cdot))\|_{L^p}^p}{|t_2 - t_1|^{1+\alpha p}} dt_1 dt_2 + 2C2^{-Np\epsilon}.
\end{align*}
$$

Now by choosing $N$ large enough, we can make the error term $2C2^{-Np\epsilon}$ as small as we want. Since the partial sum $\sum_{j=0}^N 2^{jp\alpha} \|\psi_j(\sqrt{-\Delta})(T_{n_1}\chi(t_1, \cdot) - T_{n_1}\chi(t_2, \cdot) - T_{n_2}\chi(t_1, \cdot) + T_{n_2}\chi(t_2, \cdot))\|_{L^p}$ depends only on a finite number of Fourier modes, letting $n_1, n_2 \to +\infty$, we have the term

$$t \in I \mapsto \|\psi_j(\sqrt{-\Delta})(T_{n_1}\chi(t, \cdot) - T_{n_2}\chi(t, \cdot))\|_{L^p}^p$$

goes to 0 in $W_t^{\alpha, p}(I)$ hence we conclude since the term

$$\int_{I^2} \frac{\sum_{j=0}^{N} 2^{jp\alpha} \|\psi_j(\sqrt{-\Delta})(T_{n_1}\chi(t_1, \cdot) - T_{n_1}\chi(t_2, \cdot) - T_{n_2}\chi(t_1, \cdot) + T_{n_2}\chi(t_2, \cdot))\|_{L^p}^p}{|t_2 - t_1|^{1+\alpha p}} dt_1 dt_2$$

goes to zero. $\square$

We also need to prove some compact injections in the weighted spaces since we would like to go from local to global.

**Proposition C.4 (Compact injections weighted versions).** For $\alpha \in (0, 1)$, $\varepsilon > 0$ s.t. $\alpha + \varepsilon \in (0, 1)$, we have the compact injections of the weighted spaces $C^{\alpha, \alpha-1, s} \hookrightarrow C^{\alpha+\varepsilon, \alpha-1+\varepsilon, s-\varepsilon}$ is compact.

It is important that on the r.h.s. the weight has decreased.

*Proof.* Recall that a larger scaling exponent $s$ in the weighted Sobolev norms means your distribution has better decay near the singular points. The proof is an easy consequence of the above compactness embedding just taking the weights into account. $\square$

# D Symbolic index

We collect in this appendix commonly used symbols of the article, together with their meaning and, if relevant, the page where they first occur.

<table><thead><tr><th>Symbol</th><th>Meaning</th><th>Page</th></tr></thead><tbody><tr><td>G</td><td>A compact Lie group</td><td></td></tr><tr><td>1<sub>G</sub></td><td>Identity on G</td><td></td></tr><tr><td>log</td><td>A local inverse of exp extended to all G</td><td></td></tr><tr><td>[g]</td><td>Conjugacy class of the element g ∈ G</td><td></td></tr><tr><td>Ĝ</td><td>The set of equivalence classes of irreps of G</td><td></td></tr><tr><td>λ</td><td>An equivalent class of an irrep</td><td></td></tr><tr><td>χ<sub>λ</sub></td><td>The character of λ</td><td></td></tr><tr><td>d<sub>λ</sub></td><td>The dimension of any representation in λ</td><td></td></tr><tr><td>g</td><td>Lie algebra of G</td><td></td></tr></tbody></table>
<table><thead><tr><th>Symbol</th><th>Meaning</th><th>Page</th></tr></thead><tbody><tr><td>Σ, M</td><td>A compact surface</td><td></td></tr><tr><td>σ</td><td>Smooth area form</td><td></td></tr><tr><td>∂Σ</td><td>Boundary of Σ</td><td></td></tr><tr><td>A</td><td>A g-valued 1-form on Σ</td><td></td></tr><tr><td>Tr</td><td>A trace</td><td></td></tr><tr><td>Ω<sup>1</sup>(M, g)</td><td>Space of g-valued 1-forms on M</td><td></td></tr><tr><td>Hol(A, c)</td><td>Parallel transport, or holonomy of A along the curve c</td><td>4</td></tr><tr><td>S<sub>YM</sub>(A)</td><td>Yang-Mills action of A</td><td>4</td></tr><tr><td>G</td><td>The gauge group C<sup>∞</sup>(M, G)</td><td>4</td></tr><tr><td>g · A</td><td>The gauge transform of A by g ∈ G</td><td>4</td></tr><tr><td>Z<sub>τ</sub></td><td>Segal amplitude</td><td>8</td></tr><tr><td>dA</td><td>Exterior differential of A</td><td></td></tr><tr><td>A ∧ A</td><td>Exterior product of A and A for the Lie bracket</td><td></td></tr><tr><td>ξ</td><td>A white noise</td><td></td></tr><tr><td>C<sup>∞</sup>(., *)</td><td>Smooth *-valued functions on .</td><td></td></tr><tr><td>D'(., *)</td><td>Topological dual of C<sup>∞</sup>(., *)</td><td></td></tr><tr><td>θ<sub>1</sub> → θ<sub>2</sub></td><td>The segment {(r, θ<sub>1</sub> + t); 0 ≤ t ≤ θ<sub>2</sub> - θ<sub>1</sub>} of a level set</td><td></td></tr><tr><td>od</td><td>Stratonovich differential</td><td>14</td></tr><tr><td>□(r, r', θ, θ')</td><td>The region of Σ bounded by the flowlines θ, θ' and the level sets r and r'</td><td></td></tr><tr><td>B<sup>s</sup><sub>p,q</sub></td><td>Besov regularity scale</td><td></td></tr><tr><td>W<sup>s,p</sup></td><td>Sobolev regularity scale</td><td></td></tr><tr><td>C<sup>s</sup></td><td>Hölder-Besov regularity scale</td><td></td></tr><tr><td>P(X ∈ dx), f<sub>X</sub></td><td>Probability density of X with respect to dx</td><td></td></tr><tr><td></td><td>Morse-Smale</td><td>21</td></tr><tr><td></td><td>Morse Chart</td><td>19</td></tr><tr><td></td><td>Adapted Metric</td><td>20</td></tr><tr><td>W<sup>s</sup>(a), W<sup>u</sup>(a)</td><td>Stable, Unstable manifolds of a</td><td>21</td></tr><tr><td>U<sub>a</sub></td><td>Unstable current</td><td>21</td></tr><tr><td>M(★)</td><td>Set of probability measures on ★ equipped with weak topology</td><td></td></tr></tbody></table>

## References

[1] Aizenbud, Avraham, and Dmitry Gourevitch. "Schwartz functions on Nash manifolds." International Mathematics Research Notices 2008 (2008): rnm155.

[2] G. Alberti, E. Stepanov, and D. Trevisan, Integration of nonsmooth 2-forms: from Young to Itô and Stratonovich. J. Funct. Anal., 286(2):37, 2024.

[3] M. Audin and M. Damian, Morse theory and Floer homology, Springer, 2014.

[4] Baladi, Viviane, and Masato Tsujii. "Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms." Annales de l'institut Fourier. Vol. 57. No. 1. 2007.

[5] Baladi, Viviane, and Masato Tsujii. "Dynamical determinants and spectrum for hyperbolic diffeomorphisms." Contemporary Mathematics (2008): 29-68.
[6] Bayer, Christian, and Peter K. Friz. *Cubature on Wiener space: pathwise convergence*. Applied Mathematics and Optimization 67.2 (2013): 261-278.

[7] Bony, Jean-Michel. *Interaction des singularités pour les équations aux dérivées partielles non linéaires*. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1981): 1-11.

[8] Bony, Jean-Michel. *Second micro-localization and propagation of singularities for semi-linear hyperbolic equations*. Hyperbolic equations and related topics. Academic Press, 1986. 11-49.

[9] A. Chandra and H. Singh, Rough geometric integration. Preprint, arXiv:2405.16615 [math.DG] (2024), 2024.

[10] Minjae Park, Joshua Pfeffer, Scott Sheffiel, and Pu Yu, Wilson loop expectations as sums over surfaces on the plane, Probability and Mathematical Physics, 7:1, 2026

[11] Bahouri, Hajer. Fourier analysis and nonlinear partial differential equations. Springer, 2011.

[12] T. Balaban, Recent Results in Constructing Gauge Fields, Physica 124A (1984) 79-90 79 North-Holland, Amsterdam.

[13] Baladi, Viviane. Dynamical zeta functions and dynamical determinants for hyperbolic maps. Berlin: Springer International Publishing, 2018.

[14] Baladi, Viviane. "The quest for the ultimate anisotropic Banach space." Journal of Statistical Physics 166.3 (2017): 525-557.

[15] Bonatti, Christian, Lorenzo J. Díaz, and Marcelo Viana. Dynamics beyond uniform hyperbolicity: A global geometric and probabilistic perspective. Berlin, Heidelberg: Springer Berlin Heidelberg, 2005.

[16] G. Bonthonneau, R. Chhaibi, V. Dang, G. Rivière, T. Tô, Morse-de Rham model for surface Yang-Mills measures, in preparation.

[17] G. Bonthonneau, R. Chhaibi, V. Dang, G. Rivière, T. Tô, in preparation.

[18] Breuillard, Emmanuel, Peter Friz, and Martin Huesmann. "From random walks to rough paths." Proceedings of the American Mathematical Society 137.10 (2009): 3487-3496.

[19] Bringmann, Bjoern, and Sky Cao. "A para-controlled approach to the stochastic Yang-Mills equation in two dimensions." arXiv preprint arXiv:2305.07197 (2023).

[20] Bringmann, Bjoern, and Sky Cao. "Global well-posedness of the stochastic Abelian-Higgs equations in two dimensions." arXiv preprint arXiv:2403.16878 (2024).

[21] Cao, Sky, and Sourav Chatterjee. "The Yang-Mills heat flow with random distributional initial data." Communications in Partial Differential Equations 48.2 (2023): 209-251.

[22] Cao, Sky, and Sourav Chatterjee. "A state space for 3D Euclidean Yang-Mills theories." Communications in Mathematical Physics 405.1 (2024): 3.

[23] Casselman, William, Henryk Hecht, and Dragan Milicic. "Bruhat filtrations and Whittaker vectors for real groups." Proceedings of symposia in pure mathematics. Vol. 68. Providence, RI; American Mathematical Society; 1998, 2000.

[24] Chevyrev, I. Yang-Mills Measure on the Two-Dimensional Torus as a Random Distribution. Commun. Math. Phys. 372, 1027-1058 (2019).
[25] Chevyrev, Ilya, and Hao Shen. "Invariant measure and universality of the 2D Yang-Mills Langevin dynamic." arXiv preprint arXiv:2302.12160 (2023).

[26] Chandra, A., Chevyrev, I., Hairer, M., and Shen, H. (2022). Langevin dynamic for the 2D Yang–Mills measure. Publications mathématiques de l'IHÉS, 136(1), 1-147.

[27] Chandra, A., Chevyrev, I., Hairer, M., and Shen, H. (2024). Stochastic quantisation of yang-mills-higgs in 3d. Inventiones mathematicae, 237(2), 541-696.

[28] Chevyrev, I., Garban, C. Villain Action in Lattice Gauge Theory. J Stat Phys 192, 38 (2025).
https://doi.org/10.1007/s10955-025-03420-1

[29] Dahlqvist, Antoine, and Lemoine, Thibaut. Large N limit of the Yang-Mills measure on compact surfaces II: Makeenko-Migdal equations and the planar master field. Forum of Mathematics, Sigma, Volume 13, e17, 2025.

[30] Dahlqvist, Antoine, and Lemoine, Thibaut. Large N limit of Yang-Mills partition function and Wilson loops on compact surfaces. Probability and Mathematical Physics, No. 4, 849–890, 2023.

[31] Dang, Nguyen Viet, and Estanislao Herscovich. "Renormalization of quantum field theory on Riemannian manifolds." Reviews in Mathematical Physics 31.06 (2019): 1950017.

[32] Dang, Nguyen Viet. "The extension of distributions on manifolds, a microlocal approach." Annales Henri Poincaré. Vol. 17. No. 4. Cham: Springer International Publishing, 2016.

[33] Dang, Nohra, Rivière. "Notes on rough differential equations on Lie groups", in preparation.

[34] Dang, Nguyen Viet, and Gabriel Rivière. "Spectral analysis of Morse-Smale gradient flows." Ann. Sci. ENS, Vol. 52 (2019), 1403-1458.

[35] Dang, Nguyen Viet, and Gabriel Riviere. "Pollicott-Ruelle spectrum and Witten laplacians." Journal of the European Mathematical Society 23.6 (2021): 1797-1857.

[36] Dyatlov, Semyon, and Maciej Zworski. "Dynamical zeta functions for Anosov flows via microlocal analysis." Annales scientifiques de l'École normale supérieure. Vol. 49. No. 3. 2016.

[37] B. K. Driver. YM2: continuum expectations, lattice convergence, and lassos. Comm. Math. Phys., 123(4):575–616, 1989.

[38] B. K. Driver. A functional integral approaches to the Makeenko-Migdal equations. Comm. Math. Phys., 370(1):49–116, 2019.

[39] Faure, Frédéric, and Johannes Sjöstrand. "Upper bound on the density of Ruelle resonances for Anosov flows." Communications in mathematical physics 308.2 (2011): 325-364.

[40] X. Fernique, Processus linéaires, processus généralisés, Annales de l'institut Fourier, tome 17, no 1 (1967), p. 1-92.

[41] Fine, Dana S. "Quantum Yang-Mills on a Riemann surface." Communications in mathematical physics 140 (1991): 321-338.

[42] Friz, Peter K., and Martin Hairer. A course on rough paths. Berlin: Springer International Publishing, 2020.

[43] Grieser, Daniel. "Basics of the b-calculus." Approaches to Singular Analysis: A Volume of Advances in Partial Differential Equations. Basel: Birkhäuser Basel, 2001. 30-84.
[44] Gubinelli, Massimiliano, Peter Imkeller, and Nicolas Perkowski. "Paracontrolled distributions and singular PDEs." Forum of Mathematics, Pi. Vol. 3. Cambridge University Press, 2015.

[45] Harvey, F. Reese, and H. B. Lawson. "Morse theory and Stokes' theorem." Surveys in differential geometry (2000): 259-311.

[46] Harvey, F. Reese, and H. B. Lawson. "Finite Volume Flows and Morse Theory." Annals of Math. 153(1) (2001), 1-25.

[47] Peter Hintz, *Lectures on geometric singular analysis, with applications to elliptic and hyperbolic PDE*

[48] T. Jaffard, Hölder Regularity of Distributional Volume Forms. Arxiv preprint, 2025.

[49] Kashiwara, Masaki. "The Riemann-Hilbert problem for holonomic systems." Publications of the Research Institute for Mathematical Sciences 20.2 (1984): 319-365.

[50] Laudenbach, François. "On the Thom-Smale complex." Astérisque 205 (1992): 219-233.

[51] Leoni, Giovanni. A first course in fractional Sobolev spaces. Vol. 229. American Mathematical Society, 2023.

[52] Lévy, Thierry. Yang-Mills measure on compact surfaces. American Mathematical Soc., 2003.

[53] Lévy, Thierry. "Two-dimensional Markovian holonomy fields." Astérisque 1 (2010).

[54] Lévy, Thierry. "The Master Field on the Plane." Astérisque

[55] Lévy, Thierry, and Mylène Maïda. "Central limit theorem for the heat kernel measure on the unitary group." Journal of Functional Analysis 259.12 (2010): 3163-3204.

[56] Lévy, T., Norris, J. Large Deviations for the Yang-Mills Measure on a Compact Surface. Commun. Math. Phys. 261, 405-450 (2006). https://doi.org/10.1007/s00220-005-1450-2

[57] Malgrange, Bernard. *Ideals of Differentiable Functions*, Tata Institute of Fundamental Research Studies in Mathematics, No. 3 (Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967).

[58] Martin, Jörg. "Refinements of the solution theory for singular SPDEs." (2018).

[59] Meyer, Yves. Wavelets, vibrations and scalings. No. 9. American Mathematical Soc., 1998.

[60] Migdal, A. A. "Recursion equations in gauge field theories" 30 Years of the Landau Institute - Selected Papers. Edited by KHALATNIKOV ISAAK M ET AL. Published by World Scientific Publishing Co. Pte. Ltd., 1996. ISBN 9789814317344, pp. 114-119.

[61] Migdal, A. A. "Phase transitions in gauge and spin-lattice systems." Zh. Eksp. Teor. Fiz 69 (1975): 1457.

[62] Di Nezza, Eleonora, Giampiero Palatucci, and Enrico Valdinoci. "Hitchhiker's guide to the fractional Sobolev spaces." Bulletin des sciences mathématiques 136.5 (2012): 521-573.

[63] Isao Sauzedde, Renormalised Amperean Area of Brownian Motions and Symanzik Representation of the 2D Abelian Yang-Mills-Higgs Field, arXiv preprint arXiv:2412.16781 (2024)

[64] Seeley, Robert T. "Extension of $C^\infty$ functions defined in a half space." Proceedings of the American Mathematical Society 15.4 (1964): 625-626.
[65] Sengupta, Ambar. "The Yang-Mills measure for S2." Volume 108, Issue 2, Journal of Functional Analysis, 1992, Pages 231-273, ISSN 0022-1236.

[66] Sengupta, Ambar. Gauge theory on compact surfaces. Vol. 600. American Mathematical Soc., 1997.

[67] Shen H, Smith SA, Zhu R. A new derivation of the finite N master loop equation for lattice Yang-Mills. Electronic Journal of Probability. 2024;29:1-8.

[68] Shen, H., Zhu, R. and Zhu, X., 2023. A stochastic analysis approach to lattice Yang-Mills at strong coupling. Communications in Mathematical Physics, 400(2), pp.805-851.

[69] del Teso, Félix, David Gómez-Castro, and Juan Luis Vázquez. "Estimates on translations and Taylor expansions in fractional Sobolev spaces." Nonlinear Analysis 200 (2020): 111995.

[70] Vasy, András. "A minicourse on microlocal analysis for wave propagation." Asymptotic analysis in general relativity 443 (2018): 219-374.

[71] Viana, Marcelo, and Krerley Oliveira. Foundations of ergodic theory. No. 151. Cambridge University Press, 2016.

[72] Viana, Marcelo. Lectures on Lyapunov exponents. Vol. 145. Cambridge University Press, 2014.

[73] Witten, Edward. "On quantum gauge theories in two dimensions." Communications in Mathematical Physics 141.1 (1991): 153-209.

[74] Witten, Edward. "Two dimensional gauge theories revisited." Journal of Geometry and Physics 9.4 (1992): 303-368.

[75] R. Züst, Integration of Hölder forms and currents in snowflake spaces. Calc. Var. Partial Differ. Equ., 40(1-2):99-124, 2011.