We begin with the estimation of overlap, which is easier.

As the i-th collision between particles ($q_i, \bar{q}_i$) can last only a time $\frac{\epsilon}{|v_{q_i}(\tau_i^-) - v_{\bar{q}_i}(\tau_i^-)|}$ (B.1), there can be an overlap only if there is some $\tau \in \delta\mathbb{Z} \cap [0, t]$ such that $\tau_i$ is in the interval

$$I_{\epsilon}(\tau, V_{n_K}, \nu_{[n_k-1]}) := \left[ \tau - \frac{\epsilon}{|v_{q_i}(\tau_i^-) - v_{\bar{q}_i}(\tau_i^-)|}, \tau \right]$$

Hence, the set of parameters leading to an overlap is smaller than

$$\sum_{\substack{\tau \in \delta\mathbb{Z} \cap [0,t] \\ 1 \le i \le n_k-1}} \sum_{T} \int_{\mathfrak{T}_{n_k} \times \mathfrak{G}_T^0} 1_{I_{\epsilon}(\tau, V_{n_K}, \nu_{[n_k-1]})(\tau_i)} \Lambda_T(V_{n_K}, \nu_{[n_k-1]}) d\nu_{[n_k-1]} d\tau_{[n_k-1]} M^{\otimes n_k} dV_{n_k} \\ \le \frac{t}{\delta} \frac{C^{n_k} t^{n_k-1} \theta^{(n_k-n_{k-1}-1)_+}}{n_k^{n_k}} \sum_{1 \le i \le n_k-1} \sum_{T} \int_{\mathfrak{G}_T^0} \frac{\epsilon \Lambda_T(V_{n_K}, \nu_{[n_k-1]})}{|v_{q_i}(\tau_i^-) - v_{\bar{q}_i}(\tau_i^-)|} d\nu_{[n_k-1]} M^{\otimes n_k} dV_{n_k}$$

As

$$\frac{\epsilon \Lambda_T(V_{n_K}, \nu_{[n_k-1]})}{|v_{q_i}(\tau_i^-) - v_{\bar{q}_i}(\tau_i^-)|} = \prod_{\substack{j=1 \\ j \ne i}}^{n_k-1} \left( (v_{q_j}^{\epsilon}(\tau_i) - v_{\bar{q}_j}^{\epsilon}(\tau_j)) \cdot \nu_j \right)_+$$

we can apply the same estimates as in Lemma 6.1. We conclude that the set of overlap has a measure smaller than

$$\frac{\epsilon C^{n_k} t^{n_k-1+1} \theta^{(n_k-n_{k-1}-1)_+}}{\delta n_k^{n_k}}.$$

We treat the recollision now. If the first recollision involves particles $q$ and $q'$ at time $\tau_{\text{rec}}$, we consider $\omega \subset [n_{k+2}]$ the connected components of $\{q, q'\}$ in the collision graph $\mathcal{G}^{[0, \tau_{\text{rec}}]}$. Before the time $\tau_{\text{rec}}$, the pseudotrajectory $Z_{\omega}^{\epsilon}(\tau)$ and its formal limit $Z_{\omega}(\tau)$ are close up to a translation, and using Lemma 6.3, there exists a $y_0 \in \mathbb{T}$ such that

$$\forall \tau \in [0, \tau_{\text{rec}}], |X_{\omega}^{0}(\tau) - \text{tr}_{y_0} X_{\omega}^{\epsilon}(\tau)| \le \sum_{i=1}^{n_{k+2}-1} \frac{2n_K \mathbb{V} \epsilon}{|v_{q_i}(\tau_i^-) - v_{\bar{q}_i}(\tau_i^-)|}.$$

Hence, there is a recollision if at time $\tau_{\text{rec}} \in [0, t] \setminus \{\tau_i\}_i$,

$$(B.6) \quad \exists q, q' \in [n_k - 1] \text{ such that } |x_q^0(\tau_{\text{rec}}) - x_{q'}^0(\tau_{\text{rec}})| \le \epsilon + \sum_{i=1}^{n_{k+2}-1} \frac{2n_K \mathbb{V} \epsilon}{|v_{q_i}(\tau_i^-) - v_{\bar{q}_i}(\tau_i^-)|}.$$

We can study only the limiting flow and defining an touch as "there exists a time $\tau_{\text{rec}}$ such that (B.6) is verified: we have

$$\mathcal{T}_{\underline{n}_k} \times \mathfrak{G}_T^0 \setminus \mathcal{T}_{\underline{n}_k} \times \mathfrak{G}_T^\epsilon \subset \mathcal{T}_{\underline{n}_k} \times \mathfrak{G}_T^0 \cap \{\text{at least one touch}\}.$$

The first step is to forbid the collisions which last to long. We define $\mathcal{E}_1 \subset \mathcal{T}_{\underline{n}_k} \times \mathfrak{G}_T^0$ as

$$\mathcal{E}_1 := \left\{ \forall i \le n_k - 1, |v_{q_i}(\tau_i^-) - v_{\bar{q}_i}(\tau_i^-)| \min \{1, (\tau_i - \tau_{i-1}), (\tau_{i+1} - \tau_i)\} \ge \frac{\epsilon^{1/4}}{n_k^2 \mathbb{V}} \right\}.$$

In $\mathcal{E}_1$ there is a touch if there exists a time $\tau_{\text{rec}}$ such that

$$(B.7) \qquad |x_q^0(\tau_{\text{rec}}) - x_{q'}^0(\tau_{\text{rec}})| \le 3\epsilon^{3/4},$$

and

$$1 - 1_{\mathcal{E}_1} \le \sum_{i=1}^{n_k-1} 1_{|v_{q_i}(\tau_i^-) - v_{\bar{q}_i}(\tau_i^-)| \le \frac{\epsilon c_1}{n_k^2 \mathbb{V}}} + 1_{|v_{q_i}(\tau_i^-) - v_{\bar{q}_i}(\tau_i^-)| \max\{(\tau_i - \tau_{i-1}), (\tau_{i+1} - \tau_i)\} \le \frac{\epsilon^{1/4}}{n_k^2 \mathbb{V}}}.$$

Now fix a collision tree $T$. The first touch happens at time $\tau_{\text{rec}}$ between particles $q_{\text{rec}}$ and $q'_{\text{rec}}$. There exists a collision $i_0$, two disjoint sequences of collisions $(i_j)_{j \le p}$ and $(i'_j)_{j \le p'}$ and two sequences of particles $(a_j)_{j \le p}$, $(a'_j)_{j \le p'}$ such that

* $\forall j \ge 1, i_0 < i_j$ and $i_0 < i'_j$,

* $\forall j \ge 1, a_j \in \{q_i, q'_i\} \cap \{q_{i-1}, q'_{i-1}\}$ and $a'_j \in \{q_{i'}, q'_{i'}\} \cap \{q_{i'-1}, q'_{i'-1}\}$,

* if for $j < j'$, $a_j = a_{j'}$, then for any $i \in [i_j, i_{j'}]$ such that $a_j \in \{q_i, q'_i\}$ we have $j \in \{i_j, i_{j+1}, \dots, i_{j'}\}$, and similarly for the sequences $(i'_j)_{j \le p'}, (a'_j)_{j \le p'}$,

* $a_0 = q_{i_0}, a'_0 = q'_{i_0}$ and $\{a_p, a'_{p'}\} = \{q_{\text{rec}}, q_{\text{rec}}\}$.
We have a touch if

$$ (B.8) \quad \min_{\substack{s \in [-t, t] \\ y_0 \in \mathbb{Z}^d}} \left| y_0 + \sum_{j=1}^{p} \nu_{a_j}(|\tau_{i_{j-1}}, \tau_{i_j}|(\tau_{i_j} - \tau_{i_{j-1}}) + \nu_{a_p}(|\tau_{i_p}, \tau_{\text{rec}}|)(s - \tau_{i_p}) - \sum_{j=1}^{p'} \nu_{a'_j}(|\tau'_{i'_{j-1}}, \tau'_{i'_j}|(\tau'_{i'_j} - \tau'_{i'_{j-1}}) + \nu_{a'_{p'}}(|\tau'_{i'_{p'}}, \tau_{\text{rec}}|)(s - \tau'_{i'_{p'}}) ) \right| \le \varepsilon^{3/4}. $$

As the velocities are bound by $n_k \mathbb{V}$, $y_0$ is smaller than $n_k^2 \mathbb{V}t$.

**Remark B.2.1.** Note that we can perform the previous construction if $\tau_{\text{rec}} > \tau_{i_0}$. If we study the overlap for particles of size $\varepsilon$, it is always the case. But for punctual particles, the pathology can happen before the first collision. Our proof can be adapted by taking $i_0$ such that $i_0 > i_j$ and $i_0 > i')_{j}$, making the change of variable $V_{n_k} \mapsto V_{n_k}(\tau_{i_0}^{-})$ and looking everything backwardly.

Denoting

$$ \begin{aligned} \Delta \mathbf{v}_j &:= \mathbf{v}_{a_{j+1}}(|\tau_{i_j}, \tau_{i_{j+1}}|) - \mathbf{v}_{a_j}(|\tau_{i_{j-1}}, \tau_{i_j}|), \Delta \mathbf{v}'_j &:= \mathbf{v}_{a'_{j+1}}(|\tau'_{i'_j}, \tau'_{i'_{j+1}}|) - \mathbf{v}'_{a'_j}(|\tau'_{i'_j-1}, \tau'_{i'_j}|), \\ w_0 &:= \mathbf{v}_{a_0}(\tau_{i_0}^{+}) - \mathbf{v}_{a'_0}(\tau_{i_0}^{+}), \text{ and } w_f &:= \mathbf{v}_{a_p}(|\tau_{i_p}, \tau_{\text{rec}}|) - \mathbf{v}_{a'_{p'}}(|\tau'_{i'_{p'}}, \tau_{\text{rec}}|), \end{aligned} $$

the equation (B.8) can be written as

$$ (B.9) \quad \min_{\substack{s \in [-t, t] \\ y_0 \in \mathbb{Z}^d}} \left| y_0 - \tau_{i_0} w_0 + \sum_{j=1}^{p} \Delta \mathbf{v}_j \tau_{i_j} - \sum_{j=1}^{p'} \Delta \mathbf{v}'_j \tau'_{i'_j} + sw_f \right| \le \varepsilon^{3/4} $$

We define $\mathcal{E}_2 \subset \mathcal{E}_1$ as

$$ (B.10) \quad \mathcal{E}_2 := \left\{ \forall s \in [-t, t], i \in [1, n_k], y_0 \in \mathbb{Z}^d \setminus \{0, |y_0 - s(\mathbf{v}_{q_i}(\tau_i^+) - \mathbf{v}_{q'_i}(\tau_i^+))| \ge \varepsilon^{1/4} \right\}. $$

We want to bound the complement of $\mathcal{E}_2$ in $\mathcal{E}_1$. As it is a subset of $\mathcal{E}_1$, the case $y_0 = 0$ is impossible. The velocities are bounded by $\mathbb{V}$ and we only have to test the $y_0$ such that $|y_0| \le t \mathbb{V}$. If

$$ |y_0 - s(\mathbf{v}_{q_i}(\tau_i^+) - \mathbf{v}_{q'_i}(\tau_i^+))| \le \varepsilon^{1/4}, $$

the vector $(\mathbf{v}_{q_i}(\tau_i^+) - \mathbf{v}_{q'_i}(\tau_i^+))$ has to leave in a cone $\mathcal{C}(y_0, \varepsilon^{1/4})$ of axes $y_0$ and angle $2 \arcsin \varepsilon^{1/4} / |y_0| \lesssim 2\varepsilon^{1/4}$ ($y_0$ is of length at least 1). Hence

$$ (B.11) \quad \mathbb{1}_{\text{touch}} \mathbb{1}_{\mathcal{E}_1} (1 - \mathbb{1}_{\mathcal{E}_2}) \le \sum_{\substack{y_0 \in \mathbb{Z}^d \\ |y_0| \le t \mathbb{V}}} \sum_{i=1}^{n_k-1} \mathbb{1}_{\mathbf{v}_{q_i}(\tau_i^+) - \mathbf{v}_{q'_i}(\tau_i^+) \in \mathcal{C}(y_0, \varepsilon^{1/4})}. $$

Now we place our-self in $\mathcal{E}_2$. There exists a $\ell_0$ such that $|\Delta \mathbf{v}_{\ell_0}|$ or $|\Delta \mathbf{v}'_{\ell_0}|$ is smaller than $\frac{\varepsilon^{1/4}}{2n_k t}$ (in the following, we suppose that it is $|\Delta \mathbf{v}_{\ell_0}|$). Else, by triangular inequality

$$ |w_f - w_0| = \left| \sum_{j=1}^{p} \Delta \mathbf{v}_j - \sum_{j=1}^{p'} \Delta \mathbf{v}'_j \right| \le \frac{\varepsilon^{1/4}}{3n_k t}, $$

which gives the following contradiction

$$ \exists s \in \mathbb{R}, |y_0 - (\tau_{i_0} - s)w_0| \le \varepsilon^{3/4} + \frac{2\varepsilon^{1/4}}{3}. $$

In the following, we denote $\hat{w}_f := \frac{w_f}{|w_f|}$. Equation (B.9) implies that

$$ (B.12) \quad \min_{y_0 \in \mathbb{Z}^d} \left| (y_0 - \tau_{i_0} w_0) \wedge \hat{w}_f + \sum_{j=1}^{p} \Delta \mathbf{v}_j \wedge \hat{w}_f \tau_{i_j} - \sum_{j=1}^{p'} \Delta \mathbf{v}'_j \wedge \hat{w}_f \tau'_{i'_j} \right| \le \varepsilon^{3/4}. $$

We distinguish two cases:

* If there exists a $\ell_1 \in \mathbb{N}$ such that $|\Delta \mathbf{v}_{\ell_1} \wedge \hat{w}_f| \ge \varepsilon^{1/2}/n_k$, $\tau_{i_{\ell_1}}$ has to be in an interval of length $\varepsilon^{1/4}$.

* Else, we have

$$ |w_0 \wedge \hat{w}_f| \le \sum_{j=1}^{p} |\Delta \mathbf{v}_j \wedge \hat{w}_f| + \sum_{j=1}^{p'} |\Delta \mathbf{v}'_j \wedge \hat{w}_f| \le \varepsilon^{1/2}. $$
If $y_0$ is non zero, $|y_0 \wedge w_f|$ is smaller than $2t\mathbb{V}\epsilon^{1/2}$. Hence, there exists a collision $j$, a vector $y_0 \in \mathbb{Z}^d$ with $|y_0| \le n_K t\mathbb{V}$, and a couple of particles $(q, q')$ such that

$$ \left| (v_q(\tau_j^-) - v_{q'}(\tau_j^-)) \wedge y_0 \right| \le 2t\mathbb{V}\epsilon^{1/2} $$

If $y_0 = 0$, as we are in $\mathcal{E}_1$,

$$ \left| \frac{\Delta v_{\ell_0}}{|\Delta v_{\ell_0}|} \wedge \hat{w}_f \right| \le \frac{\epsilon^{1/2}}{n_k |\Delta v_{\ell_0}|} \le \epsilon^{1/4} \quad \text{and} \quad |w_0 \wedge \hat{w}_f| \le \frac{1}{\tau_0} \left( \epsilon^{3/4} + n_k \frac{\epsilon^{1/2}}{n_k} \right) \le \epsilon^{1/2}. $$

Finally

$$ (B.13) \qquad \left| \frac{\Delta v_{\ell_0}}{|\Delta v_{\ell_0}|} \wedge w_0 \right| \le \epsilon^{1/2} + \epsilon^{1/4}. $$

Note that $\Delta v_{\ell_0}$ is equal one of the $(\pm \zeta_i(v_{q\ell_0}(\tau_{\ell_0}^-) - v_{q'\ell_0}(\tau_{\ell_0}^-), v_{\ell_0})_{i \le 4}$, where, denoting

$$ (\frac{w'}{2}, -\frac{w'}{2}, \nu') = \xi_{\alpha}((w/2, -w/2, \nu)), $$

$$ \zeta_1(w, \nu) := w, \quad \zeta_2(w, \nu) := w', \quad \zeta_3(w, \nu) := \frac{w-w'}{2} \quad \text{and} \quad \zeta_4(w, \nu) := \frac{w+w'}{2}. $$

We conclude that

$$ \begin{aligned} & \int_{\Sigma_{n_k}} d\tau_{n_k} \mathbb{1}_{\text{touch}} \\ \le & \frac{C^{n_k} t^{n_{k-1}} \theta^{(n_k-n_{k-1}-1)+}}{n_k^{n_k}} \left( \epsilon^{1/4} + \sum_i \mathbb{1}_{|v_{q_i}(\tau_i^-) - v_{q_i}(\tau_i^-)| \le \epsilon^{1/4}} + \frac{\epsilon^{1/4}}{|v_{q_i}(\tau_i^-) - v_{q_i}(\tau_i^-)|} \right. \\ & \qquad \left. + \sum_{\substack{y_0 \in \mathbb{Z}^d \\ |y_0| \le tn_k \mathbb{V}(q, q')}} \sum_i \mathbb{1}_{|v_q(\tau_j^-) - v_{q'}(\tau_j^-)| \wedge y_0} \le 2t\mathbb{V}\epsilon^{1/2}} \\ & \qquad + \sum_{i,j} \sum_{\ell=1}^4 \mathbb{1} \left| \frac{\zeta_{\ell}(v_{q_j}(\tau_j^-) - v_{q_j'}(\tau_j^-), \nu_j)}{|\zeta_{\ell}(v_{q_j}(\tau_j^-) - v_{q_j'}(\tau_j^-), \nu_j)|} \wedge (v_{q_i}(\tau_i^+) - v_{q_i'}(\tau_i^+)) \right| \le \epsilon^{1/4} \right). \end{aligned} $$

We have to integrate now with respect to $(\nu_{[n_k-1]}, V_{n_k})$. As in the proof of (6.11) we use the applications $\Xi_T^i$ defined by (6.13).

We recall that

$$ \Xi_T^{i+1} \Xi_T^i \cdots \Xi_T^1(\nu_{[n_k-1]}, V_{n_k}) = (\tilde{\nu}_{[n_k-1]}, \tilde{V}_{n_k} = V_{n_k}(\tau_i^+)) $$

and that the Jacobian of this application is 1. We can integrate with respect to $v_q(\tau_i^-) - v_q(\tau_i^-)$.

We treat now the last singularity. If we remove the edges $(q_j, q'_j)$ from $T$, either $q_j$ or $q'_j$ is not in the connected component of $\{q_i, q'_i\}$. Without loss of generality, we suppose that it is $q_j$. We denote $\omega$ the connected components of $q_j$ in $T \setminus \{(q_j, q'_j, s_j)\}$. Before the collision $j$ the particles of $\omega$ are independent of the other ones, and as before, we can construct an application of Jacobian 1

$$ \bar{\Xi} : (\nu_{[n_k-1]}, V_{n_k}) \mapsto (\bar{\nu}, V_{\omega^c}(\tau_i^+), V_{\omega}(\tau_j^-)). $$

In addition

$$ \begin{aligned} |V_{n_k}|^2 & = \frac{|V_{n_k}(\tau_i^+)|^2}{2} + \frac{|V_{n_k}(\tau_j^-)|^2}{2} \\ & \ge \frac{|V_{\omega^c}(\tau_i^+)|^2}{2} + \frac{|v_{q_j}(\tau_j^-)|^2 + |v_{q_j}(\tau_j^-)|^2}{2} + \frac{|V_{\omega \setminus \{q_j\}}(\tau_j^-)|^2}{2} \\ & \ge \frac{|V_{\omega^c}(\tau_i^+)|^2}{2} + \frac{|v_{q_j}(\tau_j^-) - v_{q_j}(\tau_j^-)|^2}{4} + \frac{|V_{\omega \setminus \{q_j\}}(\tau_j^-)|^2}{2}, \end{aligned} $$
Two particles $(q_j, q'_j)$ can touch on a time interval shorter than $\frac{\epsilon}{|v_{q_j}(\tau_j)-v_{\bar{q}_j}(\tau_j)|}$ (which is integrable). Hence, using the same strategy than in the proof of Proposition 5.2:

$$ \int_{\mathbb{T}^{r-1} \times B_r(\mathbf{V})} 1_{\mathcal{R}^{\text{mult}}(\lambda_1, \dots, \lambda_l)} \frac{e^{-\mathcal{H}_r}}{(2\pi)^{\frac{rd}{2}}} dX_{2,r} dV_r \le \frac{C_r \delta^{r-1}}{(\mu \mathbf{d})^{r-1}} \epsilon^a. $$

Summing on all the possible $(\lambda_1, \dots, \lambda_l)$ we obtain the expected result.

<figure><img src="image_14.png" /></figure>

**Proof of (8.10).** We have now to prove the Estimation (8.10):

$$ \int_{\mathbb{T}^{r-1} \times (\mathbb{R}^d)^n} 1_{\mathcal{E}_\infty} 1_{Z_r \text{ form a possible cluster}} \frac{e^{-\frac{1}{2} \mathcal{H}_r(Z_r)}}{(2\pi)^{dr/2}} dX_{2,r} dV_r \le \frac{C_r \delta^{\min\{2,r-1\}}}{(\mu \mathbf{d})^{r-1}} \epsilon^a, $$

Without loss of generality, we suppose that $1 \in \mathcal{V}$.

Fix the family $\omega_1, \dots, \omega_p$ of subset covering $[n]$, and $(\lambda_i)_{i \le p} = (\lambda_i^1, \dots, \lambda_i^{l_i})_{i \le p}$ a family of partition of $\omega_i$ such that the union of the collision graph associated to parameters $((\lambda_i^j)_{j})_i$ is connected.

We begin by fix $Z_\infty$ and sum the clustering of the particle in $[n] \setminus \mathcal{V}$

$$ \int_{\mathbb{T}^{n-1} \times (\mathbb{R}^d)^n} 1_{\mathcal{E}_\infty} 1_{Z_r \text{ form a possible cluster}} \frac{e^{-\frac{1}{2} \mathcal{H}_n(Z_n)}}{(2\pi)^{dn/2}} dX_\infty dV_\infty \le 1_{\mathcal{E}_\infty} \frac{e^{-\frac{1}{2} \mathcal{H}|\mathcal{V}|(Z_\infty)}}{(2\pi)^{d|\mathcal{V}|/2}} \frac{C_n \delta^{n-|\mathcal{V}|}}{(\mu \mathbf{d})^{|\mathcal{V}|}} \epsilon^a. $$

Then we integrate with respect to $dX_\infty\setminus\{1\} dV_\infty$.

<figure><img src="image_15.png" /></figure>

**Acknowledgment:** The authors Sergio Simonella and Raphael Winter for their numerous suggestions as well as for the misprint corrections which helped to improve the paper. He also thanks Laure Saint-Raymond for stimulating and fruitful discussions. The author acknowledges financial support from the European Union (ERC, PASTIS, Grant Agreement n°101075879).<fnref n="4" />

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