arXiv subjects
Publications and source records attributed to Claude Godrèche.
The distribution of the first positive position reached by a random walker starting at the origin is central to the analysis of extremes and records in one-dimensional random walks. In this work, we present a detailed and self-contained analytical study of this distribution for symmetric finite-range lattice walks, whose steps are drawn from a distribution supported on finitely many integers.
Building upon the knowledge of the distribution of the first positive position reached by a random walker starting from the origin, one can derive new results on the statistics of the gap between the largest and second-largest positions of the walk, and recover known ones in a more direct manner.
The distribution of the first positive position reached by a random walker starting from the origin is fundamental for understanding the statistics of extremes and records in one-dimensional random walks. We present a comprehensive study of this distribution, focusing particularly on its moments and asymptotic tail behaviour, in the case where the step distribution is continuous and symmetric, encompassing both diffusive random walks and Lévy flights.
This paper investigates various geometrical properties of interfaces of the two-dimensional voter model. Despite its simplicity, the model exhibits dual characteristics, resembling both a critical system with long-range correlations, while also showing a tendency towards order similar to the Ising-Glauber model at zero temperature. This duality is reflected in the geometrical properties of its interfaces, which are examined here from the perspective of Schramm-Loewner evolution. Recent studies have delved into the geometrical properties of these interfaces within different lattice geometries and boundary conditions. We revisit these findings, focusing on a system within a box of linear size $L$ with Dobrushin boundary conditions, where values of the spins are fixed to either $+1$ or $-1$ on two distinct halves of the boundary, in order to enforce the presence of a pinned interface with fixed endpoints (or chordal interface). We also expand the study to compare the geometrical properties of the interfaces of the voter model with those of the critical Ising model and other related models. Scaling arguments and numerical studies suggest that, while locally the chordal interface of the voter model has fractal dimension $d_{\rm f}=3/2$, corresponding to a parameter $κ=4$, it becomes straight at large scales, confirming a conjecture made by Holmes et al \cite{holmes}, and ruling out the possibility of describing the chordal interface of the voter model by SLE$_κ$, for any non zero value of $κ$. This contrasts with the critical Ising model, which is described by SLE$_3$, and whose interface fluctuations remain of order $L$, and more generally with related critical models, which are in the same universality class.
Convex records have an appealing purely geometric definition. In a sequence of $d$-dimensional data points, the $n$-th point is a convex record if it lies outside the convex hull of all preceding points. We specifically focus on the bivariate (i.e., two-dimensional) setting. For iid (independent and identically distributed) points, we establish an identity relating the mean number $\mean{R_n}$ of convex records up to time $n$ to the mean number $\mean{N_n}$ of vertices in the convex hull of the first $n$ points. By combining this identity with extensive numerical simulations, we provide a comprehensive overview of the statistics of convex records for various examples of iid data points in the plane: uniform points in the square and in the disk, Gaussian points and points with an isotropic power-law distribution. In all these cases, the mean values and variances of $N_n$ and $R_n$ grow proportionally to each other, resulting in finite limit Fano factors $F_N$ and $F_R$. We also consider planar random walks, i.e., sequences of points with iid increments. For both the Pearson walk in the continuum and the Pólya walk on a lattice, we characterise the growth of the mean number $\mean{R_n}$ of convex records and demonstrate that the ratio $R_n/\mean{R_n}$ keeps fluctuating with a universal limit distribution.
We replicate a renewal process at random times, which is equivalent to nesting two renewal processes, or considering a renewal process subject to stochastic resetting. We investigate the consequences on the statistical properties of the model of the intricate interplay between the two probability laws governing the distribution of time intervals between renewals, on the one hand, and of time intervals between resettings, on the other hand. In particular, the total number ${\mathcal N}_t$ of renewal events occurring within a specified observation time exhibits a remarkable range of behaviours, depending on the exponents characterising the power-law decays of the two probability distributions. Specifically, ${\mathcal N}_t$ can either grow linearly in time and have relatively negligible fluctuations, or grow subextensively over time while continuing to fluctuate. These behaviours highlight the dominance of the most regular process across all regions of the phase diagram. In the presence of Poissonian resetting, the statistics of ${\mathcal N}_t$ is described by a unique `dressed' renewal process, which is a deformation of the renewal process without resetting. We also discuss the relevance of the present study to first passage under restart and to continuous time random walks subject to stochastic resetting.
We consider the simple random walk (or Pólya walk) on the one-dimensional lattice subject to stochastic resetting to the origin with probability $r$ at each time step. The focus is on the joint statistics of the numbers ${\mathcal{N}}_t^{\times}$ of spontaneous returns of the walker to the origin and ${\mathcal{N}}_t^{\bullet}$ of resetting events up to some observation time $t$. These numbers are extensive in time in a strong sense: all their joint cumulants grow linearly in $t$, with explicitly computable amplitudes, and their fluctuations are described by a smooth bivariate large deviation function. A non-trivial crossover phenomenon takes place in the regime of weak resetting and late times. Remarkably, the time intervals between spontaneous returns to the origin of the reset random walk form a renewal process described in terms of a single `dressed' probability distribution. These time intervals are probabilistic copies of the first one, the `dressed' first-passage time. The present work follows a broader study, covered in a companion paper, on general nested renewal processes.
For a stochastic process reset at random times, we discuss to what extent the probabilities of some orderings of observables associated with the intervals of time between resetting events are universal, i.e., independent of the choice of the observables, and in particular, to what extent universality depends on the choice of the distribution of these intervals. For Poissonian resetting, universality relies only on a combinatorial argument and on the statistical properties of Poisson points. For a generic distribution of time intervals between resets, universality no longer holds in general.
This work is devoted to an in-depth analysis of arbitrary temperature protocols applied to the ferromagnetic Glauber-Ising chain launched from a disordered initial state and evolving in the low-temperature scaling regime. We focus our study on the density of domain walls and the reduced susceptibility. Both the inverse of the former observable and the latter one provide two independent measures of the typical size of the growing ferromagnetic domains. Their product is thus a dimensionless form factor characterising the pattern of growing ordered domains and providing a measure of the distance of the system to thermal equilibrium. We apply this framework to a variety of protocols: everlasting slow quenches, where temperature decreases continuously to zero in the limit of infinitely long times, slow quenches of finite duration, where temperature reaches zero at some long but finite quenching time, time-periodic protocols with weak and strong modulations, and the two-temperature protocol leading to the memory effect found by Kovacs.
A semi-Markov process is one that changes states in accordance with a Markov chain but takes a random amount of time between changes. We consider the generalisation to semi-Markov processes of the classical Lamperti law for the occupation time of a two-state Markov process. We provide an explicit expression in Laplace space for the distribution of an arbitrary linear combination of the occupation times in the various states of the process. We discuss several consequences of this result. In particular, we infer the limiting distribution of this quantity rescaled by time in the long-time scaling regime, as well as the finite-time corrections to its moments.
We revisit the statistics of extremes and records of symmetric random walks with stochastic resetting, extending earlier studies in several directions. We put forward a diffusive scaling regime (symmetric step length distribution with finite variance, weak resetting probability) where the maximum of the walk and the number of its records up to discrete time $n$ become asymptotically proportional to each other for single typical trajectories. Their distributions obey scaling laws ruled by a common two-parameter scaling function, interpolating between a half-Gaussian and a Gumbel law. The exact solution of the problem for the symmetric exponential step length distribution and for the simple Polya lattice walk, as well as a heuristic analysis of other distributions, allow a quantitative study of several facets of the statistics of extremes and records beyond the diffusive scaling regime.
We perform a thorough analysis of the survival probability of symmetric random walks with stochastic resetting, defined as the probability for the walker not to cross the origin up to time $n$. For continuous symmetric distributions of step lengths with either finite (random walks) or infinite variance (Lévy flights), this probability can be expressed in terms of the survival probability of the walk without resetting, given by Sparre Andersen theory. It is therefore universal, i.e., independent of the step length distribution. We analyze this survival probability at depth, deriving both exact results at finite times and asymptotic late-time results. We also investigate the case where the step length distribution is symmetric but not continuous, focussing our attention onto arithmetic distributions generating random walks on the lattice of integers. We investigate in detail the example of the simple Polya walk and propose an algebraic approach for lattice walks with a larger range.
What is the probability that a needle dropped at random on a set of points scattered on a line segment does not fall on any of them? We compute the exact scaling expression of this hole probability when the spacings between the points are independent identically distributed random variables with a power-law distribution of index less than unity, implying that the average spacing diverges. The theoretical framework for such a setting is renewal theory, to which the present study brings a new contribution. The question posed here is also related to the study of some correlation functions of simple models of statistical physics.
Renewal processes generated by a power-law distribution of intervals with tail index less than unity are genuinely non-stationary. This issue is illustrated by a critical review of the recent paper by Barma, Majumdar and Mukamel 2019 \textit{J.~Phys.~A} {\bf 52} 254001, devoted to the investigation of the properties of a specific one-dimensional equilibrium spin system with long-range interactions. We explain why discarding the non-stationarity of the process underlying the model leads to an incorrect expression of the critical spin-spin correlation function, even when the system, subjected to periodic boundary conditions, is translation invariant.
We address the theory of records for integrated random walks with finite variance. The long-time continuum limit of these walks is a non-Markov process known as the random acceleration process or the integral of Brownian motion. In this limit, the renewal structure of the record process is the cornerstone for the analysis of its statistics. We thus obtain the analytical expressions of several characteristics of the process, notably the distribution of the total duration of record runs (sequences of consecutive records), which is the continuum analogue of the number of records of the integrated random walks. This result is universal, i.e., independent of the details of the parent distribution of the step lengths.
We investigate the statistics of three kinds of records associated with planar random walks, namely diagonal, simultaneous and radial records. The mean numbers of these records grow as universal power laws of time, with respective exponents 1/4, 1/3 and 1/2. The study of diagonal and simultaneous records relies on the underlying renewal structure of the successive hitting times and locations of translated copies of a fixed target. In this sense, this work represents a two-dimensional extension of the analysis made by Feller of ladder points, i.e., records for one-dimensional random walks. This approach yields a variety of analytical asymptotic results, including the full statistics of the numbers of diagonal and simultaneous records, the joint law of the epoch and location of the current diagonal record and the angular distribution of the current simultaneous record. The sequence of radial records cannot be constructed in terms of a renewal process. In spite of this, their mean number is shown to grow with a super-universal square-root law for isotropic random walks in any spatial dimension. Their full distribution is also obtained. Higher-dimensional diagonal and simultaneous records are also briefly discussed.
We address the question of condensation and extremes for three classes of intimately related stochastic processes: (a) random allocation models and zero-range processes, (b) tied-down renewal processes, (c) free renewal processes. While for the former class the number of components of the system is fixed, for the two other classes it is a fluctuating quantity. Studies of these topics are scattered in the literature and usually dressed up in other clothing. We give a stripped-down account of the subject in the language of sums of independent random variables in order to free ourselves of the consideration of particular models and highlight the essentials. Besides giving a unified presentation of the theory, this work investigates facets so far unexplored in previous studies. Specifically, we show how the study of the class of random allocation models and zero-range processes can serve as a backdrop for the study of the two other classes of processes central to the present work -- tied-down and free renewal processes. We then present new insights on the extreme value statistics of these three classes of processes which allow a deeper understanding of the mechanism of condensation and the quantitative analysis of the fluctuations of the condensate.
We investigate how the statistics of extremes and records is affected when taking the moving average over a window of width $p$ of a sequence of independent, identically distributed random variables. An asymptotic analysis of the general case, corroborated by exact results for three distributions (exponential, uniform, power-law with unit exponent), evidences a very robust dichotomy, irrespective of the window width, between superexponential and subexponential distributions. For superexponential distributions the statistics of records is asymptotically unchanged by taking the moving average, up to interesting distribution-dependent corrections to scaling. For subexponential distributions the probability of record breaking at late times is increased by a universal factor $R_p$, depending only on the window width.