arXiv subjects

Rémy Poudevigne

Publications and source records attributed to Rémy Poudevigne.

7 recordsLinked to original sources

arXiv2026

Stochastic domination and lifts of random variables in percolation theory

Consider some matrix waiting for its coefficients to be written. For each column, sample independently a Bernoulli random variable of some parameter $p$. Seeing all this and possibly using extra randomness, Alice then chooses one spot in each column, in any way she wants. When the Bernoulli random variable of some column is equal to 1, the number 1 is written in the chosen spot. When the Bernoulli random variable of a column is 0, nothing is done on this column. We prove that, using extra randomness, it is possible for Bob to fill the empty entries with well chosen 0's and 1's so that the entries of the matrix are independent Bernoulli random variables of parameter $p$. We investigate various generalisations and variations of this problem, and use this result to revisit and generalise (nonstrict) monotonicity of the percolation threshold $p_c$ with respect to a form of graph-quotienting, namely fibrations. We also use this result to revisit the BK inequality. In a second part, which is independent of the first one, we revisit strict monotonicity of $p_c$ with respect to fibrations, a result that naturally requires more assumptions than its nonstrict counterpart. We reprove the bond-percolation case of the result of Martineau--Severo without resorting to essential enhancements, using couplings instead.

math.PR↗

arXiv2025

Existence of critical tiltings and local limits of general size-conditioned Bienaymé-Galton-Watson multitype trees

We are interested in the structure of multitype Bienaymé-Galton-Watson (BGW) trees conditioned on integer linear combinations of the numbers of vertices of given types. We show that, under regularity assumptions on the offspring distributions, it is always possible to find a critical BGW tree having the same conditional distribution. This allows us to prove the existence of local limits for noncritical BGW trees, under a large variety of conditionings. Our proof is based on geometric considerations on the set of the so-called exponential tiltings of a family of offspring distributions.

math.PR↗

arXiv2025

A random polymer approach to the weak disorder phase of the vertex reinforced jump process

In this paper, we study the transient phase of the Vertex Reinforced Jump Process (VRJP) in dimension $d\geq 3$. In Sabot, Zeng (2019), the authors introduce a positive martingale and show that the VRJP is recurrent if and only if that martingale converges to $0$. On $\mathbb{Z}^d$, $d\ge 3$, with constant conductances $W$, it can be shown that there is a critical value $0 W_c(\mathbb{Z}^d)$. On the other hand, the VRJP martingale can be interpreted as the partition function of a non-directed polymer with a very specific $1$-dependent random potential. In this paper, we focus on the question of the $L^p$ integrability of the VRJP martingale, which is related to the (diffusive) behavior of the VRJP. First, taking inspiration from the work of Junk (2022) for directed polymers in $\mathbb{Z}^{1+d}$, we prove that on the half-space $\mathbb{H}_d$ of $\mathbb{Z}^d$, for all $W>W_c(\mathbb{H}_d)$ there is some $δ>0$ such that the VRJP martingale is in $L^{1+δ}$. Second, we prove that, in dimension $d\geq 4$, the VRJP martingale is in $L^{p}$ for all $p>1$ above the ``slab critical point'' $W_c^{\mathrm{slab}} (\mathbb{Z}^d) = \lim_{m\to\infty} W_c(\mathbb{Z}^{d-1} \times \{-m,\ldots,m\})$. We also propose some related conjectures.

math.PR↗

arXiv2024

$\mathbb{H}^{2|2}$-model and Vertex-Reinforced Jump Process on Regular Trees: Infinite-Order Transition and an Intermediate Phase

We explore the supercritical phase of the vertex-reinforced jump process (VRJP) and the $\mathbb{H}^{2|2}$-model on rooted regular trees. The VRJP is a random walk, which is more likely to jump to vertices on which it has previously spent a lot of time. The $\mathbb{H}^{2|2}$-model is a supersymmetric lattice spin model, originally introduced as a toy model for the Anderson transition. On infinite rooted regular trees, the VRJP undergoes a recurrence/transience transition controlled by an inverse temperature parameter $β> 0$. Approaching the critical point from the transient regime, $β\searrow β_{\mathrm{c}}$, we show that the expected total time spent at the starting vertex diverges as $\sim \exp(c/\sqrt{β- β_{\mathrm{c}}})$. Moreover, on large finite trees we show that the VRJP exhibits an additional intermediate regime for parameter values $β_{\mathrm{c}} < β< β_{\mathrm{c}}^{\mathrm{erg}}$. In this regime, despite being transient in infinite volume, the VRJP on finite trees spends an unusually long time at the starting vertex with high probability. We provide analogous results for correlation functions of the $\mathbb{H}^{2|2}$-model. Our proofs rely on the application of branching random walk methods to a horospherical marginal of the $\mathbb{H}^{2|2}$-model.

math.PR↗

arXiv2022

Macroscopic cycles for the interchange and quantum Heisenberg models on random regular graphs

The interchange process is a random permutation model that was introduced as a way to study the quantum Heisenberg model. For this model, progress had been made on some specific graphs: trees, the hypercube, the Hamming graph, the complete graph and the two block graph. Here we show that for large enough parameters, both the interchange process and the quantum Heisenberg model have macroscopic clusters on random d-regular graphs. Such a result was only known for the complete graph and the two blocks graph.

math.PR↗

arXiv2019

Monotonicity and phase transition for the VRJP and the ERRW

The vertex-reinforced jump process (VRJP), introduced by Davis and Volkov, is a continuous-time process that tends to come-back to already visited vertices. It is closely linked to the edge-reinforced random walk (ERRW) introduced by Coppersmith and Diaconis in 1986 which is more likely to cross edges it has already crossed. On $\mathbb{Z}^d$ for $d\geq 3$, both models where shown to be recurrent for small enough initial weights (by Sabot, Tarrès(2015) and Angel,Crawford,Kozma(2014)) and transient for large enough initial weights (by Disertori,Sabot,Tarrès(2015) and Sabot,Tarrès(2015)). We show through a coupling of the VRJP for different weights that the VRJP (and the ERRW) exhibits some monotonicity. In particular, we show that increasing the initial weights of the VRJP and the ERRW makes them more transient which means that the recurrence/transience phase transition is necessarily unique. Furthermore, by making the weights go to infinity, we show that the recurrence of the ERRW and the VRJP is implied by the recurrence of a random walk in deterministic electrical network.

math.PR↗

arXiv2019

Random walk in random environment and their time-reversed counterpart

The random walk in Dirichlet environment is a random walk in random environment where the transition probabilities are independent Dirichlet random variables. This random walk exhibits a property of statistical invariance by time-reversal which leads to several results. More precisely, a time-reversed random walk in Dirichlet environment (with null divergence) is also a random walk in random environment where the transition probabilities are independent Dirichlet random variables with different parameters. We show that on all graphs that satisfy a few weak assumptions, a random walk in random environment with independent transition probabilities and such that the transition probabilities of the time-reversed random walk in random environment are also independent is a random walk in Dirichlet environment.

math.PR↗