arXiv subjects

Francis Comets

Publications and source records attributed to Francis Comets.

At least 19 recordsLinked to original sources

arXiv2023

Brownian Polymers in Poissonian Environment: a survey

We consider a space-time continuous directed polymer in random environment. The path is Brownian and the medium is Poissonian. We review many results obtained in the last decade, and also we present new ones. In this fundamental setup, we can make use of fine formulas and strong tools from stochastic analysis for Gaussian or Poisson measure, together with martingale techniques. These notes cover the matter of a course presented during the Jean-Morlet chair 2017 of CIRM "Random Structures in Statistical Mechanics and Mathematical Physics" in Marseille.

math.PR↗

arXiv2022

Limit of the environment viewed from Sinaï's walk

For Sinaï's walk (X_k) we show that the empirical measure of the environment seen from the particle (\bar\w_k) converges in law to some random measure S. This limit measure is explicitly given in terms of the infinite valley, which construction goes back to Golosov. As a consequence an "in law" ergodic theorem holds for additive functionals of (\bar\w_k) . When the limit in this "in law" ergodic theorem is deterministic, it holds in probability. This allows some extensions to the recurrent case of the ballistic "environment's method" dating back to Kozlov and Molchanov. In particular, we show an LLN and a mixed CLT for the sums sum_{k=1}^nf(ΔX_k), where f is bounded and depending on the steps ΔX_k:=X_{k+1}-X_k.

math.PR↗

arXiv2022

Generalizations of forest fires with ignition at origin

We study generalizations of the Forest Fire model introduced in [van den Berg, J., and Járai, A. A. "On the asymptotic density in a one-dimensional self-organized critical forest-fire model". Comm. Math. Phys. 253 (2005)] and [Volkov, Stanislav. "Forest fires on $\mathbb{Z}_+$ with ignition only at 0". ALEA 6 (2009)] by allowing the rates at which the tree grow to depend on their location, introducing long-range burning, as well as continuous-space generalization of the model. We establish that in all the models in consideration the time required to reach site at distance $x$ from the origin is of order at most $(\log x)^{(\log 2)^{-1}+δ}$ for any $δ>0$.

math.PR↗

arXiv2022

Scaling limit of the heavy-tailed ballistic deposition model with $p$-sticking

Ballistic deposition is a classical model for interface growth in which unit blocks fall down vertically at random on the different sites of $\mathbb{Z}$ and stick to the interface at the first point of contact, causing it to grow. We consider an alternative version of this model in which the blocks have random heights which are i.i.d. with a heavy (right) tail, and where each block sticks to the interface at the first point of contact with probability $p$ (otherwise, it falls straight down until it lands on a block belonging to the interface). We study scaling limits of the resulting interface for the different values of $p$ and show that there is a phase transition as $p$ goes from $1$ to $0$.

math.PR↗

arXiv2021

Rate of escape of conditioned Brownian motion

We study the norm of the two-dimensional Brownian motion conditioned to stay outside the unit disk at all times. By conditioning the process is changed from barely recurrent to slightly transient. We obtain sharp results on the rate of escape to infinity of the process of future minima: (i) we find an integral test on the function $g$ so that the future minima process drops beyond the barrier $\exp \{ \ln t \times g(\ln \ln t)\}$ at arbitrary large times; (ii) we show that the future minima process exceeds $K \sqrt{ t \times \ln \ln \ln t}$ at arbitrary large times with probability 0 [resp., 1] if $K$ is larger [resp., smaller] than some positive constant. For this, we introduce a renewal structure attached to record times and values. Additional results are given for the long time behavior of the norm.

math.PR↗

arXiv2021

Space-time fluctuation of the Kardar-Parisi-Zhang equation in $d\geq 3$ and the Gaussian free field

We study the solution $h_\varepsilon$ of the Kardar-Parisi-Zhang (KPZ) equation for $d \geq 3$: $$ \frac{\partial}{\partial t} h_{\varepsilon} = \frac12 Δh_{\varepsilon} + \bigg[\frac12 |\nabla h_\varepsilon |^2 - C_\varepsilon\bigg]+ β\varepsilon^{\frac{d-2}2} ξ_{\varepsilon} $$ with $h_\varepsilon(0,x)=0$. Here $ξ_\varepsilon=ξ\star ϕ_\varepsilon$ is a spatially smoothened (at scale $\varepsilon$) Gaussian space-time white noise and $C_\varepsilon$ is a divergent constant as $\varepsilon\to 0$. When the disorder $β$ is sufficiently small and $\varepsilon\to 0$, $h_\varepsilon(t,x)- \mathfrak h^{\mathrm{st}}_{\varepsilon}(t,x)\to 0$ in probability where $\mathfrak h^{\mathrm{st}}_{\varepsilon}(t,x)$ is the {\emph stationary solution} of the KPZ equation - more precisely, $\mathfrak h^{\mathrm{st}}_{\varepsilon}$solves the above equation with a random initial condition (that is independent of the driving noise $ξ$) and its law is constant in $(\varepsilon,t,x)$. In the present article we quantify the rate of the above convergence in this regime and show that the fluctuation {\emph about} the stationary solution $$ (\varepsilon^{1-\frac d2} [h_\varepsilon(t,x) - \mathfrak h^{\mathrm{st}}_{\varepsilon}(t,x)])_{x,t} $$ converges pointwise (with finite dimensional distributions in space and time) to a Gaussian free field (GFF) evolved by the deterministic heat equation. We also identify the fluctuations {\it of} the stationary solution itself and show that the rescaled averages $\int_{\mathbb R^d} {\mathrm d} x φ(x) \varepsilon^{1-\frac d2} [\mathfrak h^{\mathrm{st}}_{\varepsilon}(t,x)- \mathbb E(\mathfrak h^{\mathrm{st}}_{\varepsilon}(t,x))]$ converge to that of the {\emph stationary solution} of the stochastic heat equation with additive noise, but with (random) {\emph GFF marginals} (instead of flat initial condition).

math.PR↗

arXiv2021

Limiting results for the free energy of directed polymers in random environment with unbounded jumps

We study asymptotics of the free energy for the directed polymer in random environment. The polymer is allowed to make unbounded jumps and the environment is given by Bernoulli variables. We first establish the existence and continuity of the free energy including the negative infinity value of the coupling constant $β$. Our proof of existence at $β=-\infty$ differs from existing ones in that it avoids the direct use of subadditivity. Secondly, we identify the asymptotics of the free energy at $β=-\infty$ in the limit of the success probability of the Bernoulli variables tending to one. It is described by using the so-called time constant of a certain directed first passage percolation. Our proof relies on a certain continuity property of the time constant, which is of independent interest.

math.PR↗

arXiv2019

Random walks avoiding their convex hull with a finite memory

Fix integers $d \geq 2$ and $k\geq d-1$. Consider a random walk $X_0, X_1, \ldots$ in $\mathbb{R}^d$ in which, given $X_0, X_1, \ldots, X_n$ ($n \geq k$), the next step $X_{n+1}$ is uniformly distributed on the unit ball centred at $X_n$, but conditioned that the line segment from $X_n$ to $X_{n+1}$ intersects the convex hull of $\{0, X_{n-k}, \ldots, X_n\}$ only at $X_n$. For $k = \infty$ this is a version of the model introduced by Angel et al., which is conjectured to be ballistic, i.e., to have a limiting speed and a limiting direction. We establish ballisticity for the finite-$k$ model, and comment on some open problems. In the case where $d=2$ and $k=1$, we obtain the limiting speed explicitly: it is $8/(9π^2)$.

math.PR↗

arXiv2019

Fluctuation and Rate of Convergence for the Stochastic Heat Equation in Weak Disorder

We consider the stochastic heat equation on $\mathbb R^d$ with multiplicative space-time white noise noise smoothed in space. For $d\geq 3$ and small noise intensity, the solution is known to converge to a strictly positive random variable as the smoothing parameter vanishes. In this regime, we study the rate of convergence and show that the pointwise fluctuations of the smoothened solutions as well as that of the underlying martingale of the Brownian directed polymer converge to a Gaussian limit.

math.PR↗

arXiv2019

Two-dimensional Brownian random interlacements

We introduce the model of two-dimensional continuous random interlacements, which is constructed using the Brownian trajectories conditioned on not hitting a fixed set (usually, a disk). This model yields the local picture of Wiener sausage on the torus around a late point. As such, it can be seen as a continuous analogue of discrete two-dimensional random interlacements [Comets, Popov, Vachkovskaia, 2016]. At the same time, one can view it as (restricted) Brownian loops through infinity. We establish a number of results analogous to these of [Comets, Popov, Vachkovskaia, 2016; Comets, Popov, 2016], as well as the results specific to the continuous case.

math.PR↗

arXiv2019

Renormalizing the Kardar-Parisi-Zhang equation in $d\geq 3$ in weak disorder

We study Kardar-Parisi-Zhang equation in spatial dimension 3 or larger driven by a Gaussian space-time white noise with a small convolution in space. When the noise intensity is small, it is known that the solutions converge to a random limit as the smoothing parameter is turned off. We identify this limit, in the case of general initial conditions ranging from flat to droplet. We provide strong approximations of the solution which obey exactly the limit law. We prove that this limit has sub-Gaussian lower tails, implying existence of all negative (and positive) moments.

math.PR↗

arXiv2018

Continuum limit of random matrix products in statistical mechanics of disordered systems

We consider a particular weak disorder limit ("continuum limit") of matrix products that arise in the analysis of disordered statistical mechanics systems, with a particular focus on random transfer matrices. The limit system is a diffusion model for which the leading Lyapunov exponent can be expressed explicitly in terms of modified Bessel functions, a formula that appears in the physical literature on these disordered systems. We provide an analysis of the diffusion system as well as of the link with the matrix products. We then apply the results to the framework considered by Derrida and Hilhorst [J. Phys. A (1983)], which deals in particular with the strong interaction limit for disordered Ising model in one dimension and that identifies a singular behavior of the Lyapunov exponent (of the transfer matrix), and to the two dimensional Ising model with columnar disorder (McCoy-Wu model). We show that the continuum limit sharply captures the Derrida and Hilhorst singularity. Moreover we revisit the analysis by McCoy and Wu [Phys. Rev. 1968] and remark that it can be interpreted in terms of the continuum limit approximation. We provide a mathematical analysis of the continuum approximation of the free energy of the McCoy-Wu model, clarifying the prediction (by McCoy and Wu) that, in this approximation, the free energy of the two dimensional Ising model with columnar disorder is $C^\infty$ but not analytic at the critical temperature.

math-ph↗

arXiv2018

Random polymers on the complete graph

Consider directed polymers in a random environment on the complete graph of size $N$. This model can be formulated as a product of i.i.d. $N\times N$ random matrices and its large time asymptotics is captured by Lyapunov exponents and the Furstenberg measure. We detail this correspondence, derive the long-time limit of the model and obtain a co-variant distribution for the polymer path. Next, we observe that the model becomes exactly solvable when the disorder variables are located on edges of the complete graph and follow a totally asymmetric stable law of index $α\in (0,1)$. Then, a certain notion of mean height of the polymer behaves like a random walk and we show that the height function is distributed around this mean according to an explicit law. Large $N$ asymptotics can be taken in this setting, for instance, for the free energy of the system and for the invariant law of the polymer height with a shift. Moreover, we give some perturbative results for environments which are close to the totally asymmetric stable laws.

math.PR↗

arXiv2017

The vacant set of two-dimensional critical random interlacement is infinite

For the model of two-dimensional random interlacements in the critical regime (i.e., $α=1$), we prove that the vacant set is a.s.\ infinite, thus solving an open problem from arXiv:1502.03470. Also, we prove that the entrance measure of simple random walk on annular domains has certain regularity properties; this result is useful when dealing with soft local times for excursion processes.

math.PR↗

arXiv2016

Rate of convergence for polymers in a weak disorder

We consider directed polymers in random environment on the lattice Z d at small inverse temperature and dimension d $\ge$ 3. Then, the normalized partition function W n is a regular martingale with limit W. We prove that n (d--2)/4 (W n -- W)/W n converges in distribution to a Gaussian law. Both the polynomial rate of convergence and the scaling with the martingale W n are different from those for polymers on trees.

math.PR↗

arXiv2016

Constrained information transmission on Erdös-Rényi graphs

We model the transmission of information of a message on the Erdös-Rény random graph with parameters $(n,p)$ and limited resources. The vertices of the graph represent servers that may broadcast a message at random. Each server has a random emission capital that decreases by one at each emission. We examine two natural dynamics: in the first dynamics, an informed server performs its attempts, then checks at each of them if the corresponding edge is open or not; in the second dynamics the informed server knows a priori who are its neighbors, and it performs all its attempts on its actual neighbors in the graph. In each case, we obtain first and second order asymptotics (law of large numbers and central limit theorem), when $n\to \infty$ and $p$ is fixed, for the final proportion of informed servers.

math.PR↗

arXiv2016

Localization Transition for Polymers in Poissonian Medium

We study a model of directed polymers in random environment in dimension $1+d$, given by a Brownian motion in a Poissonian potential. We study the effect of the density and the strength of inhomogeneities, respectively the intensity parameter $ν$ of the Poisson field and the temperature inverse $β$. Our results are: (i) fine information on the phase diagram, with quantitative estimates on the critical curve; (ii) pathwise localization at low temperature and/or large density; (iii) complete localization in a favourite corridor for large $νβ^2$ and bounded $β$.

math.PR↗

arXiv2015

Two-dimensional random interlacements and late points for random walks

We define the model of two-dimensional random interlacements using simple random walk trajectories conditioned on never hitting the origin, and then obtain some properties of this model. Also, for random walk on a large torus conditioned on not hitting the origin up to some time proportional to the mean cover time, we show that the law of the vacant set around the origin is close to that of random interlacements at the corresponding level. Thus, this new model provides a way to understand the structure of the set of late points of the covering process from a microscopic point of view.

math.PR↗