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This paper investigates L\'evy walks with random velocities, extending classical models beyond constant speed assumptions. We derive scaling limits, demonstrating that diffusion depends on interplay between heavy-tailed duration and…

Probability · Mathematics 2026-04-28 Hubert Woszczek , Marek A. Teuerle , Agnieszka Wyłomańska

We study the first-passage properties of a jump process with constant drift where jump amplitudes and inter-arrival times follow arbitrary light-tailed distributions with smooth densities. Using a mapping to an effective discrete-time…

Statistical Mechanics · Physics 2026-03-25 Ivan N. Burenev

In this paper we observe the frog model, an infinite system of interacting random walks, on Z with an asymmetric underlying random walk. Under the assumption of transience with a fixed frog distribution, we construct an explicit formula for…

Probability · Mathematics 2015-02-11 Arka P. Ghosh , Steven Noren , Alexander Roitershtein

We show aging of Glauber-type dynamics on the random energy model, in the sense that we obtain the scaling limits of the clock process and of the age process. The latter encodes the Gibbs weight of the configuration occupied by the…

Probability · Mathematics 2013-03-21 Pierre Mathieu , Jean-Christophe Mourrat

We study K-processes, which are Markov processes in a denumerable state space, all of whose elements are stable, with the exception of a single state, starting from which the process enters finite sets of stable states with uniform…

Probability · Mathematics 2007-05-23 Luiz Renato Fontes , Pierre Mathieu

Let $\mathcal X=\{\mathcal X_t:\, t\geq0,\, \mathcal X_0=0\}$ be a mean zero $\beta$-stable random walk on $\mathbb{Z}$ with inhomogeneous jump rates $\{\tau_i^{-1}: i\in\mathbb{Z}\}$, with $\beta\in(1,2]$ and $\{\tau_i: i\in\mathbb{Z}\}$ a…

Probability · Mathematics 2021-04-02 W. Barreto-Souza , L. R. G. Fontes

There are two key paradigms for non-equilibrium dynamics: on the one hand, aging towards an equilibrium state that cannot be reached on reasonable timescales; on the other, external driving that can lead to non-equilibrium steady states. We…

Statistical Mechanics · Physics 2024-11-06 Diego Tapias , Charles Marteau , Fabián Aguirre-López , Peter Sollich

Aging has become the paradigm to describe dynamical behavior of glassy systems, and in particular spin glasses. Trap models have been introduced as simple caricatures of effective dynamics of such systems. In this Letter we show that in a…

Disordered Systems and Neural Networks · Physics 2008-04-14 G. Ben Arous , A. Bovier , J. Cerny

This paper considers limit theorems associated with subgraph counts in the age-dependent random connection model. First, we identify regimes where the count of sub-trees converges weakly to a stable random variable under suitable…

Probability · Mathematics 2024-09-10 Christian Hirsch , Takashi Owada

We consider a time dependent trap externally manipulated in such a way that one of its bound states is brought into an instant contact with the continuum threshold, and then down again. It is shown that, in the limit of slow evolution, the…

Quantum Physics · Physics 2016-07-20 D. Sokolovski , M. Pons

We study K-processes, which are Markov processes in a denumerable state space, all of whose elements are stable, with the exception of a single state, starting from which the process enters finite sets of stable states with uniform…

Probability · Mathematics 2008-08-27 L. R. G. Fontes , P. Mathieu

We study in details the dynamics of the one dimensional symmetric trap model, via a real-space renormalization procedure which becomes exact in the limit of zero temperature. In this limit, the diffusion front in each sample consists in two…

Condensed Matter · Physics 2009-11-10 Cecile Monthus

We consider the dynamics of the disordered trap model, which is known to be completely out-of-equilibrium and to present strong localization effects in its aging phase. We are interested into the influence of an external force, when it is…

Condensed Matter · Physics 2009-11-10 Cecile Monthus

We review recent theoretical progress on glassy dynamics, with special emphasis on the importance and universality of the ``aging regime'', which is relevant to many experimental situations. The three main subjects which we address are: (i)…

Disordered Systems and Neural Networks · Physics 2008-02-03 Jean-Philippe Bouchaud , Leticia F. Cugliandolo , Jorge Kurchan , Marc Mezard

We consider trap models on Z^d, namely continuous time Markov jump process on Z^d with embedded chain given by a generic discrete time random walk, and whose mean waiting time at x is given by tau_x, with tau = (tau_x, x in Z^d) a family of…

Probability · Mathematics 2017-05-17 Luiz Renato Fontes , Pierre Mathieu

We study the limiting behavior of continuous time trawl processes which are defined using an infinitely divisible random measure of a time dependent set. In this way one is able to define separately the marginal distribution and the…

Probability · Mathematics 2017-08-10 Danijel Grahovac , Nikolai N. Leonenko , Murad S. Taqqu

A superprocess limit for an interacting birth-death particle system modelling a population with trait and physical age-structures is established. Traits of newborn offspring are inherited from the parents except when mutations occur, while…

Probability · Mathematics 2011-11-29 Sylvie Méléard , Viet Chi Tran

Trap models are intuitively appealing and often solvable models of glassy dynamics. In particular, they have been used to study aging and the resulting out-of-equilibrium fluctuation-dissipation relations between correlations and response…

Disordered Systems and Neural Networks · Physics 2007-05-23 Peter Sollich

The relaxation dynamics of zero range process (ZRP) has always been an interesting problem. In this study, we set up the relationship between ZRP and traps model, and investigate the slow dynamics of ZRP in the framework of traps model.…

Statistical Mechanics · Physics 2015-06-04 Kai Qi , Ming Tang , Aixiang Cui , Yan Fu

We analyze the coherent dynamics of excitons in three dimensional topologically disordered networks with traps. If the interactions between the nodes of the network are long ranged, i.e., algebraically decaying as a function of the distance…

Quantum Physics · Physics 2010-02-04 Oliver Muelken , Alexander Blumen