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Olivier Zindy

Publications and source records attributed to Olivier Zindy.

15 recordsLinked to original sources
arXiv2026

The branching random walk in a uniform magnetic field : magnetization concentration and overlap distributions

Adding a uniform external magnetic field to a mean-field spin-glass model usually requires a new analysis specific to the model. The disordered system we consider corresponds to the Gaussian binary branching random walk (BRW) - in the spirit of Derrida and Spohn [21] and studied from the statistical-physics point of view by Jagannath [22] - and we prove that this is not the case : a single elementary observation - that the resulting Hamiltonian is still a BRW, now with independent, but non-identically distributed, displacements - allows us to use the available results for general BRW. Combining classical and recent results on general BRW (Biggins [3], Chauvin and Rouault [13], Mallein [25]), one obtains an essentially complete picture of the model in an external magnetic field : the ground state, the free energy, the one-step replica symmetry breaking (1-RSB) transition, and the limiting genealogical overlap distribution with Poisson-Dirichlet statistics for the Gibbs weights. We then prove a strong concentration result for the magnetization under the Gibbs measure at low temperature, giving its explicit optimal value. It turns out that this one-replica statement is not enough to control the classical Ising overlap between two independently sampled configurations : an elementary counterexample (Remark 4.1) shows that concentration of each replica's magnetization does not, by itself, determine their joint correlation. We address this by developing a two-replica large-deviation argument - resting on the classical method of types and the subadditivity of Shannon entropy, and taking the form of a uniform Chernoff bound over the joint empirical type of a pair of configurations, matched against a two-replica concentration estimate - which we use to obtain the distribution of a second (Ising) hypercube-type overlap.

math.PR↗
arXiv2025

Branching Brownian motion versus Random Energy Model in the supercritical phase: overlap distribution and temperature susceptibility

In comparison with Derrida's REM, we investigate the influence of the so-called decoration processes arising in the limiting extremal processes of numerous log-correlated Gaussian fields. In particular, we focus on the branching Brownian motion and two specific quantities from statistical physics in the vicinity of the critical temperature. The first one is the two-temperature overlap, whose behavior at criticality is smoothened by the decoration process - unlike the one-temperature overlap which is identical - and the second one is the temperature susceptibility, as introduced by Sales and Bouchaud, which is strictly larger in the presence of decorations and diverges, close to the critical temperature, at the same speed as for the REM but with a different multiplicative constant. We also study some general decorated cases in order to highlight the fact that the BBM has a critical behavior in some sense to be made precise.

math.PR↗
arXiv2020

Two-temperatures overlap distribution for the 2D discrete Gaussian free field

In this paper, we prove absence of temperature chaos for the two-dimensional discrete Gaussian free field using the convergence of the full extremal process, which has been obtained recently by Biskup and Louidor. This means that the overlap of two points chosen under Gibbs measures at different temperatures has a nontrivial distribution. Whereas this distribution is the same as for the random energy model when the two points are sampled at the same temperature, we point out here that they are different when temperatures are distinct: more precisely, we prove that the mean overlap of two points chosen under Gibbs measures at different temperatures for the DGFF is strictly smaller than the REM's one. Therefore, although neither of these models exhibits temperature chaos, one could say that the DGFF is more chaotic in temperature than the REM.

math.PR↗
arXiv2019

Universality and Sharpness in Absorbing-State Phase Transitions

We consider the Activated Random Walk model in any dimension with any sleep rate and jump distribution and ergodic initial state. We show that the stabilization properties depend only on the average density of particles, regardless of how they are initially located on the lattice.

math.PR↗
arXiv2014

Poisson-Dirichlet statistics for the extremes of a log-correlated Gaussian field

We study the statistics of the extremes of a discrete Gaussian field with logarithmic correlations at the level of the Gibbs measure. The model is defined on the periodic interval $[0,1]$, and its correlation structure is nonhierarchical. It is based on a model introduced by Bacry and Muzy [Comm. Math. Phys. 236 (2003) 449-475] (see also Barral and Mandelbrot [Probab. Theory Related Fields 124 (2002) 409-430]), and is similar to the logarithmic Random Energy Model studied by Carpentier and Le Doussal [Phys. Rev. E (3) 63 (2001) 026110] and more recently by Fyodorov and Bouchaud [J. Phys. A 41 (2008) 372001]. At low temperature, it is shown that the normalized covariance of two points sampled from the Gibbs measure is either $0$ or $1$. This is used to prove that the joint distribution of the Gibbs weights converges in a suitable sense to that of a Poisson-Dirichlet variable. In particular, this proves a conjecture of Carpentier and Le Doussal that the statistics of the extremes of the log-correlated field behave as those of i.i.d. Gaussian variables and of branching Brownian motion at the level of the Gibbs measure. The method of proof is robust and is adaptable to other log-correlated Gaussian fields.

math.PR↗
arXiv2013

The precise tail behavior of the total progeny of a killed branching random walk

Consider a branching random walk on the real line with a killing barrier at zero: starting from a nonnegative point, particles reproduce and move independently, but are killed when they touch the negative half-line. The population of the killed branching random walk dies out almost surely in both critical and subcritical cases, where by subcritical case we mean that the rightmost particle of the branching random walk without killing has a negative speed, and by critical case, when this speed is zero. We investigate the total progeny of the killed branching random walk and give their precise tail distribution both in the critical and subcritical cases, which solves an open problem of Aldous [Power laws and killed branching random walks, http://www.stat.berkeley.edu/~aldous/Research/OP/brw.html].

math.PR↗
arXiv2013

Poisson-Dirichlet Statistics for the extremes of the two-dimensional discrete Gaussian Free Field

In a previous paper, the authors introduced an approach to prove that the statistics of the extremes of a log-correlated Gaussian field converge to a Poisson-Dirichlet variable at the level of the Gibbs measure at low temperature and under suitable test functions. The method is based on showing that the model admits a one-step replica symmetry breaking in spin glass terminology. This implies Poisson-Dirichlet statistics by general spin glass arguments. In this note, this approach is used to prove Poisson-Dirichlet statistics for the two-dimensional discrete Gaussian free field, where boundary effects demand a more delicate analysis.

math.PR↗
arXiv2013

Quenched limits for the fluctuations of transient random walks in random environment on Z

We consider transient nearest-neighbor random walks in random environment on Z. For a set of environments whose probability is converging to 1 as time goes to infinity, we describe the fluctuations of the hitting time of a level n, around its mean, in terms of an explicit function of the environment. Moreover, their limiting law is described using a Poisson point process whose intensity is computed. This result can be considered as the quenched analog of the classical result of Kesten, Kozlov and Spitzer [Compositio Math. 30 (1975) 145-168].

math.PR↗
arXiv2010

Stable fluctuations for ballistic random walks in random environment on Z

We consider transient random walks in random environment on Z in the positive speed (ballistic) and critical zero speed regimes. A classical result of Kesten, Kozlov and Spitzer proves that the hitting time of level $n$, after proper centering and normalization, converges to a completely asymmetric stable distribution, but does not describe its scale parameter. Following a previous article by three of the authors, where the (non-critical) zero speed case was dealt with, we give a new proof of this result in the subdiffusive case that provides a complete description of the limit law. The case of Dirichlet environment turns out to be remarkably explicit.

math.PR↗
arXiv2009

Limit laws for transient random walks in random environment on $\z$

We consider transient random walks in random environment on $\z$ with zero asymptotic speed. A classical result of Kesten, Kozlov and Spitzer says that the hitting time of the level $n$ converges in law, after a proper normalization, towards a positive stable law, but they do not obtain a description of its parameter. A different proof of this result is presented, that leads to a complete characterization of this stable law. The case of Dirichlet environment turns out to be remarkably explicit.

math.PR↗
arXiv2008

Scaling limit and aging for directed trap models

We consider one-dimensional directed trap models and suppose that the trapping times are heavy-tailed. We obtain the inverse of a stable subordinator as scaling limit and prove an aging phenomenon expressed in terms of the generalized arcsine law. These results confirm the status of universality described by Ben Arous and Černý for a large class of graphs.

math.PR↗
arXiv2008

A probabilistic representation of constants in Kesten's renewal theorem

The aims of this paper are twofold. Firstly, we derive some probabilistic representation for the constant which appears in the one-dimensional case of Kesten's renewal theorem. Secondly, we estimate the tail of some related random variable which plays an essential role in the description of the stable limit law of one-dimensional transient sub-ballistic random walks in random environment.

math.PR↗
arXiv2007

Upper limits of Sinai's walk in random scenery

We consider Sinai's walk in i.i.d. random scenery and focus our attention on a conjecture of Révész \cite{r05} concerning the upper limits of Sinai's walk in random scenery when the scenery is bounded from above. A close study of the competition between the concentration property for Sinai's walk and negative values for the scenery enables us to prove that the conjecture is true if the scenery has "thin" negative tails and is false otherwise.

math.PR↗
arXiv2007

A weakness in strong localization for Sinai's walk

Sinai's walk is a recurrent one-dimensional nearest-neighbor random walk in random environment. It is known for a phenomenon of strong localization, namely, the walk spends almost all time at or near the bottom of deep valleys of the potential. Our main result shows a weakness of this localization phenomenon: with probability one, the zones where the walk stays for the most time can be far away from the sites where the walk spends the most time. In particular, this gives a negative answer to a problem of Erdős and Révész [Mathematical Structures--Computational Mathematics--Mathematical Modelling 2 (1984) 152--157], originally formulated for the usual homogeneous random walk.

math.PR↗