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Pierre Tarrès

Publications and source records attributed to Pierre Tarrès.

At least 19 recordsLinked to original sources
arXiv2025

Once-Reinforced and Self-Interacting Random Walks beyond exchangeability

We present the first rigorous quantitative analysis of once-reinforced random walks (ORRW) on general graphs, based on a novel change of measure formula.~This enables us to prove large deviations estimates for the range of the walk to have cardinality of the order $N^{d/(d+2)}$ in dimension larger than or equal than two. We also prove that ORRW is transient on all non-amenable graphs for small reinforcement.~Moreover, we study the shape of oriented ORRW on euclidean lattices. We also provide a new approach to the study of general self-interacting random walk, which we apply to random walk in random environment, reinforced processes on oriented graphs, including the directed ORRW.

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arXiv2024

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 Tarrès (2020). It can be seen as the continuous-time counterpart to the *-Edge-Reinforced Random Walk (*-ERRW), see Bacallado (2011) and Bacallado, Sabot and Tarrès (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 Schrödinger 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 Schrödinger 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 Schrödinger operator and the associated identities are key-features of the *-VRJP.

math.PR↗
arXiv2024

The *-Vertex Reinforced Jump Process II: random Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger operator and the associated identities are key-features of the *-VRJP.

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arXiv2023

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.

math.PR↗
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.

math.PR↗
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

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.

math.PR↗
arXiv2019

Random interlacements for vertex-reinforced jump processes

We introduce random interlacements for transient vertex-reinforced jump processes on a general graph $G$. Using increasing finite subgraphs $G_n$ of $G$ with wired boundary conditions, we show convergence of the vertex-reinforced jump process on $G_n$ observed in a finite window to the random interlacement observed in the same window.

math.PR↗
arXiv2019

Convergence of vertex-reinforced jump processes to an extension of the supersymmetric hyperbolic nonlinear sigma model

In this paper, we define an extension of the supersymmetric hyperbolic nonlinear sigma model introduced by Zirnbauer. We show that it arises as a weak joint limit of a time-changed version introduced by Sabot and Tarrès of the vertex-reinforced jump process. It describes the asymptotics of rescaled crossing numbers, rescaled fluctuations of local times, asymptotic local times on a logarithmic scale, endpoints of paths, and last exit trees.

math.PR↗
arXiv2016

Reinforcement learning in social networks

We propose a model of network formation based on reinforcement learning, which can be seen as a generalization as the one proposed by Skyrms for signaling games. On a discrete graph, whose vertices represent individuals, at any time step each of them picks one of its neighbors with a probability proportional to their past number of communications; independently, Nature chooses, with an independent identical distribution in time, which ones are allowed to communicate. Communications occur when any two neighbors mutually pick each other and are both allowed by Nature to communicate. Our results generalize the ones obtained by Hu, Skyrms and Tarrès (2011). We prove that, up to an error term, the expected rate of communications increases in average, and thus a.s. converges. If we define the limit graph as the non-oriented subgraph on which edges are pairs of vertices communicating with a positive asymptotic rate, then, for stable configurations, within which every vertex is connected to at least another one, the connected components of this limit graph are star-shaped and satisfy a certain balance condition. Conversely, given any stable equilibrium $q$ whose associated graph satisfies that property, the occupation measure converges with positive probability to a stable equilibrium in a neighborhood of $q$ with the same limit graph.

math.PR↗
arXiv2016

The Vertex Reinforced Jump Process and a Random Schrödinger operator on finite graphs

We introduce a new exponential family of probability distributions, which can be viewed as a multivariate generalization of the Inverse Gaussian distribution. Considered as the potential of a random Schrödinger operator, this exponential family is related to the random field that gives the mixing measure of the Vertex Reinforced Jump Process (VRJP), and hence to the mixing measure of the Edge Reinforced Random Walk (ERRW), the so-called magic formula. In particular, it yields by direct computation the value of the normalizing constants of these mixing measures, which solves a question raised by Diaconis. The results of this paper are instrumental in [Sabot-Zeng,2015], where several properties of the VRJP and the ERRW are proved, in particular a functional central limit theorem in transient regimes, and recurrence of the 2-dimensional ERRW.

math.PR↗
arXiv2015

Inverting Ray-Knight identity

We provide a short proof of the Ray-Knight second generalized Theorem, using a martingale which can be seen (on the positive quadrant) as the Radon-Nikodym derivative of the reversed vertex-reinforced jump process measure with respect to the Markov jump process with the same conductances. Next we show that a variant of this process provides an inversion of that Ray-Knight identity. We give a similar result for the Ray-Knight first generalized Theorem.

math.PR↗
arXiv2014

Transience of Edge-Reinforced Random Walk

We show transience of the edge-reinforced random walk (ERRW) for small reinforcement in dimension d greater than 2. This proves the existence of a phase transition between recurrent and transient behavior, thus solving an open problem stated by Diaconis in 1986. The argument adapts the proof of quasi-diffusive behavior of the SuSy hyperbolic model for fixed conductances by Disertori, Spencer and Zirnbauer [CMP 2010], using the representation of ERRW as a mixture of vertex-reinforced jump processes (VRJP) with independent gamma conductances, and the interpretation of the limit law of VRJP as a supersymmetric (SuSy) hyperbolic sigma model developed by Sabot and Tarrès in [JEMS 2014].

math.PR↗
arXiv2013

Online Learning as Stochastic Approximation of Regularization Paths

In this paper, an online learning algorithm is proposed as sequential stochastic approximation of a regularization path converging to the regression function in reproducing kernel Hilbert spaces (RKHSs). We show that it is possible to produce the best known strong (RKHS norm) convergence rate of batch learning, through a careful choice of the gain or step size sequences, depending on regularity assumptions on the regression function. The corresponding weak (mean square distance) convergence rate is optimal in the sense that it reaches the minimax and individual lower rates in the literature. In both cases we deduce almost sure convergence, using Bernstein-type inequalities for martingales in Hilbert spaces. To achieve this we develop a bias-variance decomposition similar to the batch learning setting; the bias consists in the approximation and drift errors along the regularization path, which display the same rates of convergence, and the variance arises from the sample error analysed as a reverse martingale difference sequence. The rates above are obtained by an optimal trade-off between the bias and the variance.

math.PR↗
arXiv2012

Diffusivity bounds for 1D Brownian polymers

We study the asymptotic behavior of a self-interacting one-dimensional Brownian polymer first introduced by Durrett and Rogers [Probab. Theory Related Fields 92 (1992) 337--349]. The polymer describes a stochastic process with a drift which is a certain average of its local time. We show that a smeared out version of the local time function as viewed from the actual position of the process is a Markov process in a suitably chosen function space, and that this process has a Gaussian stationary measure. As a first consequence, this enables us to partially prove a conjecture about the law of large numbers for the end-to-end displacement of the polymer formulated in Durrett and Rogers [Probab. Theory Related Fields 92 (1992) 337--349]. Next we give upper and lower bounds for the variance of the process under the stationary measure, in terms of the qualitative infrared behavior of the interaction function. In particular, we show that in the locally self-repelling case (when the process is essentially pushed by the negative gradient of its own local time) the process is super-diffusive.

math.PR↗
arXiv2012

On ergodic two-armed bandits

A device has two arms with unknown deterministic payoffs and the aim is to asymptotically identify the best one without spending too much time on the other. The Narendra algorithm offers a stochastic procedure to this end. We show under weak ergodic assumptions on these deterministic payoffs that the procedure eventually chooses the best arm (i.e., with greatest Cesaro limit) with probability one for appropriate step sequences of the algorithm. In the case of i.i.d. payoffs, this implies a "quenched" version of the "annealed" result of Lamberton, Pagès and Tarrès [Ann. Appl. Probab. 14 (2004) 1424--1454] by the law of iterated logarithm, thus generalizing it. More precisely, if $(η_{\ell,i})_{i\in \mathbb {N}}\in\{0,1\}^{\mathbb {N}}$, $\ell\in\{A,B\}$, are the deterministic reward sequences we would get if we played at time $i$, we obtain infallibility with the same assumption on nonincreasing step sequences on the payoffs as in Lamberton, Pagès and Tarrès [Ann. Appl. Probab. 14 (2004) 1424--1454], replacing the i.i.d. assumption by the hypothesis that the empirical averages $\sum_{i=1}^nη_{A,i}/n$ and $\sum_{i=1}^nη_{B,i}/n$ converge, as $n$ tends to infinity, respectively, to $θ_A$ and $θ_B$, with rate at least $1/(\log n)^{1+\varepsilon}$, for some $\varepsilon >0$. We also show a fallibility result, that is, convergence with positive probability to the choice of the wrong arm, which implies the corresponding result of Lamberton, Pagès and Tarrès [Ann. Appl. Probab. 14 (2004) 1424--1454] in the i.i.d. case.

math.PR↗
arXiv2012

Dynamics of vertex-reinforced random walks

We generalize a result from Volkov [Ann. Probab. 29 (2001) 66--91] and prove that, on a large class of locally finite connected graphs of bounded degree $(G,\sim)$ and symmetric reinforcement matrices $a=(a_{i,j})_{i,j\in G}$, the vertex-reinforced random walk (VRRW) eventually localizes with positive probability on subsets which consist of a complete $d$-partite subgraph with possible loops plus its outer boundary. We first show that, in general, any stable equilibrium of a linear symmetric replicator dynamics with positive payoffs on a graph $G$ satisfies the property that its support is a complete $d$-partite subgraph of $G$ with possible loops, for some $d\ge1$. This result is used here for the study of VRRWs, but also applies to other contexts such as evolutionary models in population genetics and game theory. Next we generalize the result of Pemantle [Probab. Theory Related Fields 92 (1992) 117--136] and Bena\"{ı}m [Ann. Probab. 25 (1997) 361--392] relating the asymptotic behavior of the VRRW to replicator dynamics. This enables us to conclude that, given any neighborhood of a strictly stable equilibrium with support $S$, the following event occurs with positive probability: the walk localizes on $S\cup\partial S$ (where $\partial S$ is the outer boundary of $S$) and the density of occupation of the VRRW converges, with polynomial rate, to a strictly stable equilibrium in this neighborhood.

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