arXiv subjects

Christophe Sabot

Publications and source records attributed to Christophe Sabot.

At least 19 recordsLinked to original sources

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 $\delta>0$ such that the VRJP martingale is in $L^{1+\delta}$. 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.

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arXiv2025

Invariant measure for the process viewed from the particle for 2D random walks in Dirichlet environment

In this paper, we consider random walks in Dirichlet random environment (RWDE) on $\mathbb{Z}^2$. We prove that, if the RWDE is recurrent (which is strongly conjectured when the weights are symmetric), then there does not exist any invariant measure for the process viewed from the particle which is absolutely continuous with respect to the static law of the environment. Besides, if the walk is directional transient and under condition $\mathbf{(T')}$, we prove that there exists such an invariant probability measure if the trapping parameter verifies $\kappa > 1$ or after acceleration of the process by a local function of the environment. This gives strong credit to a conjectural classification of cases of existence or non-existence of the invariant measure for two dimensional RWDE. The proof is based on a new identity, stated on general finite graphs, which is inspired by the representation of the $\star$-VRJP, a non-reversible generalization of the Vertex reinforced Jump Process, in terms of random Schr\"odinger operators. In the case of RWDE on 1D graph, the previous identity entails also a discrete analogue of the Matsumoto-Yor property for Brownian motion.

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arXiv2024

Recurrence and transience of the critical random walk snake in random conductances

In this paper we study the recurrence and transience of the $\mathbb{Z}^d$-valued branching random walk in random environment indexed by a critical Bienaym\'e-Galton-Watson tree, conditioned to survive. The environment is made either of random conductances or of random traps on each vertex. We show that when the offspring distribution is non degenerate with a finite third moment and the environment satisfies some suitable technical assumptions, then the process is recurrent up to dimension four, and transient otherwise. The proof is based on a truncated second moment method, which only requires to have good estimates on the quenched Green's function.

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arXiv2024

The *-Vertex Reinforced Jump Process II: random Schr\"odinger representation

In this paper we continue the analysis, initiated in the paper *-VRJP I, of the *-Vertex Reinforced Jump Process (*-VRJP), which is a non reversible generalization of the Vertex Reinforced Jump Process (VRJP). More precisely, we give a representation of the *-VRJP in terms of a random Schr\"odinger operator. The corresponding representation for the classical VRJP has proved to be very useful in the understanding of its asymptotic behavior. Several new phenomena and difficulties appear in this non reversible case due to the non exchangeability of the *-VRJP, which becomes exchangeable only after some randomization of the initial local times as proved in the companion paper. The construction is based on several new and rather remarkable identities between integrals on the space of *-symmetric and *-antisymmetric functions on vertices. We give a description of the randomized *-VRJP in terms of that random Schr\"odinger operator, which allow us to prove the representation of the randomized *-VRJP as mixture of Markov jump processes in a different and more analytic way. Similarly as for the VRJP, we think that the representation by a random Schr\"odinger operator and the associated identities are key-features of the *-VRJP.

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arXiv2023

A multi-dimensional version of Lamperti's relation and the Matsumoto-Yor processes

This paper presents a multidimensional extension of the Matsumoto-Yor properties related to exponential functionals of drifted Brownian motion. The extension involves the interaction of geometric Brownian motions which are indexed by the vertices of a finite weighted graph, and the random potential associated with the Vertex Reinforced Jump process on this graph. We prove in this context a counterpart of Lamperti's transformation, of the Markov property of the Matsumoto-Yor process and of the intertwining relation.

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arXiv2021

Inverting the Ray-Knight identity on the line

Using a divergent Bass-Burdzy flow we construct a self-repelling one-dimensional diffusion. Heuristically, it can be interpreted as a solution to an SDE with a singular drift involving a derivative of the local time. We show that this self-repelling diffusion inverts the second Ray-Knight identity on the line. The proof goes through an approximation by a self-repelling jump processes that has been previously shown by the authors to invert the Ray-Knight identity in the discrete.

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arXiv2021

The *-Edge-Reinforced Random Walk

We define a linearly reinforced process called the *-Edge-Reinforced Random Walk (*-ERRW ) which can be seen as a Yaglom reversible, hence non-reversible, extension of the Edge-Reinforced Random Walk (ERRW) introduced by Coppersmith and Diaconis in 1986. This family of processes also generalizes the r-dependent ERRW introduced by Bacallado (2009). Under some assumptions on the initial weights, the *-ERRW is partially exchangeable in the sense of Diaconis and Freedman (1980), and thus it is a random walk in a random environment. The main result of the paper gives the explicit expression of the mixing law, hence extending the "magic formula" of Coppersmith and Diaconis from the case of mixtures of reversible Markov chains to the case of mixtures of Yaglom reversible Markov chains.

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arXiv2021

The *-Vertex-Reinforced Jump Process

We investigate the non-reversible generalization of the Vertex-Reinforced Jump Process (VRJP), called the *-Vertex-Reinforced Jump Process (*-VRJP) and introduced by Bacallado, Sabot and Tarr\`es (2020). It can be seen as the continuous-time counterpart to the *-Edge-Reinforced Random Walk (*-ERRW), see Bacallado (2011) and Bacallado, Sabot and Tarr\`es (2020), which is itself a non-reversible, and in fact Yaglom reversible, generalization of the original ERRW introduced by Coppersmith and Diaconis (1986). In contrast to the classical VRJP, the *-VRJP is not exchangeable after time-change, which leads to several difficulties and new phenomena. Firstly, we show that with some appropriate randomization of the initial local time, it becomes partially exchangeable after time-change. We provide a representation of the "randomized" *-VRJP as a mixture of Yaglom reversible Markov jump processes with an explicit mixing measure, and we prove that the non-randomized *-VRJP can be written as a mixture of conditioned Markov processes. Secondly, we give a representation of the *-VRJP in terms of a random Schr\"odinger operator. The corresponding representation for the classical VRJP has proved to be very useful in the understanding of its asymptotic behavior. The construction is based on several new and rather remarkable identities between integrals on the space of *-symmetric and *-antisymmetric functions on vertices. We give a description of the randomized *-VRJP in terms of that random Schr\"odinger operator, which allows us to prove the representation of the randomized *-VRJP as a mixture of Markov jump processes in a different and more analytic manner. Similarly as for the VRJP, we think that the representation by a random Schr\"odinger operator and the associated identities are key-features of the *-VRJP.

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arXiv2020

Random walks in random hypergeometric environment

We consider one-dependent random walks on $\mathbb{Z}^d$ in random hypergeometric environment for $d\ge 3$. These are memory-one walks in a large class of environments parameterized by positive weights on directed edges and on pairs of directed edges which includes the class of Dirichlet environments as a special case. We show that the walk is a.s. transient for any choice of the parameters, and moreover that the return time has some finite positive moment. We then give a characterization for the existence of an invariant measure for the process from the point of view of the walker which is absolutely continuous with respect to the initial distribution on the environment in terms of a function $κ$ of the initial weights. These results generalize [Sab11] and [Sab13] on random walks in Dirichlet environment. It turns out that $κ$ coincides with the corresponding parameter in the Dirichlet case, and so in particular the existence of such invariant measures is independent of the weights on pairs of directed edges, and determined solely by the weights on directed edges.

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arXiv2020

A multi-dimensional version of Lamperti's relation and the Matsumoto-Yor opposite drift theorem

A classic result on the 1-dimensional Brownian motion shows that conditionally on its first hitting time of 0, it has the distribution of a 3-dimensional Bessel bridge. By applying a certain time-change to this result, Matsumoto and Yor showed a theorem giving a relation between Brownian motions with opposite drifts. The relevant time change is the one appearing in Lamperti's relation. Sabot and Zeng showed that a family of Brownian motions with interacting drifts, conditioned on the vector of hitting times of 0, also has the distribution of independent 3-dimensional Bessel bridges. Moreover, the distribution of these hitting times is related to a random potential that appears in the study of the vertex-reinforced jump process. The aim of this paper is to prove a multivariate version of the Matsumoto-Yor opposite drift theorem, by applying a Lamperti-type time change to the previous family of interacting Brownian motions. Difficulties arise since the time change progresses at different speeds on different coordinates.

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arXiv2019

Fine mesh limit of the VRJP in dimension one and Bass-Burdzy flow

We introduce a continuous space limit of the Vertex Reinforced Jump Process (VRJP) in dimension one, which we call Linearly Reinforced Motion (LRM) on $\R$. It is constructed out of a convergent Bass-Burdzy flow. The proof goes through the representation of the VRJP as a mixture of Markov jump processes. As a by-product this gives a representation in terms of a mixture of diffusions of the LRM and of the Bass-Burdzy flow itself. We also show that our continuous space limit can be obtained out of the Edge Reinforced Random Walk (ERRW), since the ERRW and the VRJP are known to be closely related. Compared to the discrete space processes, the LRM has an additional symmetry in the initial local times (initial occupation profile): changing them amounts to a deterministic change of the space and time scales.

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arXiv2019

Polynomial localization of the 2D-Vertex Reinforced Jump Process

We prove polynomial decay of the mixing field of the Vertex Reinforced Jump Process (VRJP) on $\Bbb{Z}^2$ with bounded conductances. Using [17] we deduce that the VRJP on $\Bbb{Z}^2$ with any constant conductances is almost surely recurrent. It gives a counterpart of the result of Merkl, Rolles [14] and Sabot, Zeng [17] for the 2-dimensional Edge Reinforced Random Walk.

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arXiv2019

Inverting the coupling of the signed Gausssian free field with a loop soup

Lupu introduced a coupling between a random walk loop-soup and a Gaussian free field, where the sign of the field is constant on each cluster of loops. This coupling is a signed version of isomorphism theorems relating the square of the GFF to the occupation field of Markovian trajectories. His construction starts with a loop-soup, and by adding additional randomness samples a GFF out of it. In this article we provide the inverse construction: starting from a signed free field and using a self-interacting random walk related to this field, we construct a random walk loop-soup. Our construction relies on the previous work by Sabot and Tarrès, which inverts the coupling from the square of the GFF rather than the signed GFF itself. As a consequence, we also deduce an inversion of the coupling between the random current and the FK-Ising random cluster models introduced by Lupu and Werner.

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arXiv2018

A random Schrödinger operator associated with the Vertex Reinforced Jump Process on infinite graphs

This paper concerns the Vertex reinforced jump process (VRJP), the Edge reinforced random walk (ERRW) and their link with a random Schrödinger operator. On infinite graphs, we define a 1-dependent random potential $β$ extending that defined in [20] on finite graphs, and consider its associated random Schrödinger operator $H_β$. We construct a random function $ψ$ as a limit of martingales, such that $ψ=0$ when the VRJP is recurrent, and $ψ$ is a positive generalized eigenfunction of the random Schrödinger operator with eigenvalue $0$, when the VRJP is transient. Then we prove a representation of the VRJP on infinite graphs as a mixture of Markov jump processes involving the function $ψ$, the Green function of the random Schrödinger operator and an independent Gamma random variable. On ${\mathbb Z}^d$, we deduce from this representation a zero-one law for recurrence or transience of the VRJP and the ERRW, and a functional central limit theorem for the VRJP and the ERRW at weak reinforcement in dimension $d\ge 3$, using estimates of [10,8]. Finally, we deduce recurrence of the ERRW in dimension $ d=2$ for any initial constant weights (using the estimates of Merkl and Rolles, [15,17]), thus giving a full answer to the old question of Diaconis. We also raise some questions on the links between recurrence/transience of the VRJP and localization/delocalization of the random Schrödinger operator $H_β$.

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arXiv2018

Hitting times of interacting drifted Brownian motions and the vertex reinforced jump process

Consider a negatively drifted one dimensional Brownian motion starting at positive initial position, its first hitting time to 0 has the inverse Gaussian law. Moreover, conditionally on this hitting time, the Brownian motion up to that time has the law of a 3- dimensional Bessel bridge. In this paper, we give a generalization of this result to a family of Brownian motions with interacting drifts, indexed by the vertices of a conductance network. The hitting times are equal in law to the inverse of a random potential that appears in the analysis of a self-interacting process called the Vertex Reinforced Jump Process ([17, 18]). These Brownian motions with interacting drifts have remarkable properties with respect to restriction and conditioning, showing hidden Markov properties. This family of processes are closely related to the martingale that plays a crucial role in the analysis of the vertex reinforced jump process and edge reinforced random walk ([18]) on infinite graphs.

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arXiv2017

Velocity estimates for symmetric random walks at low ballistic disorder

We derive asymptotic estimates for the velocity of random walks in random environments which are perturbations of the simple symmetric random walk but have a small local drift in a given direction. Our estimates complement previous results presented by Sznitman and are in the spirit of expansions obtained by Sabot.

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arXiv2016

Sharp ellipticity conditions for ballistic behavior of random walks in random environment

We sharpen ellipticity criteria for random walks in i.i.d. random environments introduced by Campos and Ram\'ırez which ensure ballistic behavior. Furthermore, we construct new examples of random environments for which the walk satisfies the polynomial ballisticity criteria of Berger, Drewitz and Ram\'ırez. As a corollary, we can exhibit a new range of values for the parameters of Dirichlet random environments in dimension $d=2$ under which the corresponding random walk is ballistic.

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arXiv2016

Random walks in Dirichlet environment: an overview

Random Walks in Dirichlet Environment (RWDE) correspond to Random Walks in Random Environment (RWRE) on $\Bbb{Z}^d$ where the transition probabilities are i.i.d. at each site with a Dirichlet distribution. Hence, the model is parametrized by a family of positive weights $(α_i)_{i=1, \ldots, 2d}$, one for each direction of $\Bbb{Z}^d$. In this case, the annealed law is that of a reinforced random walk, with linear reinforcement on directed edges. RWDE have a remarkable property of statistical invariance by time reversal from which can be inferred several properties that are still inaccessible for general environments, such as the equivalence of static and dynamic points of view and a description of the directionally transient and ballistic regimes. In this paper we give a state of the art on this model and several sketches of proofs presenting the core of the arguments. We also present new computation of the large deviation rate function for one dimensional RWDE.

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