Critical constants for recurrence of random walks on $G$-spaces

Anna Erschler

Displaying similar documents to “Critical constants for recurrence of random walks on $G$ -spaces”

Semidirected random polymers: Strong disorder and localization

Nikolaos Zygouras (2010)

Actes des rencontres du CIRM

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Semi-directed, random polymers can be modeled by a simple random walk on $Z^{d}$ in a random potential - ${(λ + β ω (x))}_{x \in Z^{d}}$ , where $λ > 0$ , $β > 0$ and ${(ω (x))}_{x \in Z^{d}}$ is a collection of i.i.d., nonnegative random variables. We identify situations where the annealed and quenched costs, that the polymer pays to perform long crossings are different. In these situations we show that the polymer exhibits localization.

Aging and quenched localization for one-dimensional random walks in random environment in the sub-ballistic regime

Nathanaël Enriquez, Christophe Sabot, Olivier Zindy (2009)

Bulletin de la Société Mathématique de France

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We consider transient one-dimensional random walks in a random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of “valleys“ of height $log t$ . In the quenched setting, we also sharply estimate the distribution of the walk at time $t$ .

Excited against the tide: a random walk with competing drifts

Mark Holmes (2012)

Annales de l'I.H.P. Probabilités et statistiques

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We study excited random walks in i.i.d. random cookie environments in high dimensions, where the $k$ th cookie at a site determines the transition probabilities (to the left and right) for the $k$ th departure from that site. We show that in high dimensions, when the expected right drift of the first cookie is sufficiently large, the velocity is strictly positive, regardless of the strengths and signs of subsequent cookies. Under additional conditions on the cookie environment, we show that...

Slowdown estimates and central limit theorem for random walks in random environment

Alain-Sol Sznitman (2000)

Journal of the European Mathematical Society

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This work is concerned with asymptotic properties of multi-dimensional random walks in random environment. Under Kalikow’s condition, we show a central limit theorem for random walks in random environment on $ℤ^{d}$ , when $d > 2$ . We also derive tail estimates on the probability of slowdowns. These latter estimates are of special interest due to the natural interplay between slowdowns and the presence of traps in the medium. The tail behavior of the renewal time constructed in [25] plays an important...

A Littlewood-Paley-Stein estimate on graphs and groups

Nick Dungey (2008)

Studia Mathematica

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We establish the boundedness in $L^{q}$ spaces, 1 < q ≤ 2, of a “vertical” Littlewood-Paley-Stein operator associated with a reversible random walk on a graph. This result extends to certain non-reversible random walks, including centered random walks on any finitely generated discrete group.

On the existence and asymptotic behavior of the random solutions of the random integral equation with advancing argument

Henryk Gacki (1996)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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1. Introduction Random Integral Equations play a significant role in characterizing of many biological and engineering problems [4,5,6,7]. We present here new existence theorems for a class of integral equations with advancing argument. Our method is based on the notion of a measure of noncompactness in Banach spaces and the fixed point theorem of Darbo type. We shall deal with random integral equation with advancing argument $x (t, ω) = h (t, ω) + \int^{t + δ (t)} ₀ k (t, τ, ω) f (τ, x_{τ} (ω)) d τ$ , (t,ω) ∈ R⁺ × Ω, (1) where (i) (Ω,A,P) is a complete probability...

Long time behavior of random walks on abelian groups

Alexander Bendikov, Barbara Bobikau (2010)

Colloquium Mathematicae

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Let be a locally compact non-compact metric group. Assuming that is abelian we construct symmetric aperiodic random walks on with probabilities $n \mapsto ℙ (S_{2 n} \in V)$ of return to any neighborhood V of the neutral element decaying at infinity almost as fast as the exponential function n ↦ exp(-n). We also show that for some discrete groups , the decay of the function $n \mapsto ℙ (S_{2 n} \in V)$ can be made as slow as possible by choosing appropriate aperiodic random walks Sₙ on .

Ballistic regime for random walks in random environment with unbounded jumps and Knudsen billiards

Francis Comets, Serguei Popov (2012)

Annales de l'I.H.P. Probabilités et statistiques

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We consider a random walk in a stationary ergodic environment in $ℤ$ , with unbounded jumps. In addition to uniform ellipticity and a bound on the tails of the possible jumps, we assume a condition of strong transience to the right which implies that there are no “traps.” We prove the law of large numbers with positive speed, as well as the ergodicity of the environment seen from the particle. Then, we consider Knudsen stochastic billiard with a drift in a random tube in $ℝ^{d}$ , $d \geq 3$ , which serves...

Slowdown estimates for ballistic random walk in random environment

Noam Berger (2012)

Journal of the European Mathematical Society

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We consider models of random walk in uniformly elliptic i.i.d. random environment in dimension greater than or equal to 4, satisfying a condition slightly weaker than the ballisticity condition $(T^{'})$ . We show that for every $ϵ > 0$ and $n$ large enough, the annealed probability of linear slowdown is bounded from above by $exp (- {(log n)}^{d - ϵ})$ . This bound almost matches the known lower bound of $exp (- C {(log n)}^{d})$ , and signiﬁcantly improves previously known upper bounds. As a corollary we provide almost sharp estimates for the quenched...

Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model

Christophe Sabot, Pierre Tarrès (2015)

Journal of the European Mathematical Society

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Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986 [8], is a random process which takes values in the vertex set of a graph $G$ and is more likely to cross edges it has visited before. We show that it can be represented in terms of a vertex-reinforced jump process (VRJP) with independent gamma conductances; the VRJP was conceived by Werner and first studied by Davis and Volkov [10, 11], and is a continuous-time process favouring sites with more local time....

On the limiting velocity of random walks in mixing random environment

Xiaoqin Guo (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider random walks in strong-mixing random Gibbsian environments in $ℤ^{d}$ , $d \geq 2$ . Based on regeneration arguments, we will first provide an alternative proof of Rassoul-Agha’s conditional law of large numbers (CLLN) for mixing environment ( (2005) 36–44). Then, using coupling techniques, we show that there is at most one nonzero limiting velocity in high dimensions ( $d \geq 5$ ).

Further results on laws of large numbers for uncertain random variables

Feng Hu, Xiaoting Fu, Ziyi Qu, Zhaojun Zong (2023)

Kybernetika

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The uncertainty theory was founded by Baoding Liu to characterize uncertainty information represented by humans. Basing on uncertainty theory, Yuhan Liu created chance theory to describe the complex phenomenon, in which human uncertainty and random phenomenon coexist. In this paper, our aim is to derive some laws of large numbers (LLNs) for uncertain random variables. The first theorem proved the Etemadi type LLN for uncertain random variables being functions of pairwise independent...

Weak quenched limiting distributions for transient one-dimensional random walk in a random environment

Jonathon Peterson, Gennady Samorodnitsky (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We consider a one-dimensional, transient random walk in a random i.i.d. environment. The asymptotic behaviour of such random walk depends to a large extent on a crucial parameter $κ g t; 0$ that determines the fluctuations of the process. When $0 l t; κ l t; 2$ , the averaged distributions of the hitting times of the random walk converge to a $κ$ -stable distribution. However, it was shown recently that in this case there does not exist a quenched limiting distribution of the hitting times. That is, it is not true...

Perturbing transient random walk in a random environment with cookies of maximal strength

Elisabeth Bauernschubert (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We consider a left-transient random walk in a random environment on $ℤ$ that will be disturbed by cookies inducing a drift to the right of strength 1. The number of cookies per site is i.i.d. and independent of the environment. Criteria for recurrence and transience of the random walk are obtained. For this purpose we use subcritical branching processes in random environments with immigration and formulate criteria for recurrence and transience for these processes.

Random walk in random environment with asymptotically zero perturbation

M.V. Menshikov, Andrew R. Wade (2006)

Journal of the European Mathematical Society

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We give criteria for ergodicity, transience and null-recurrence for the random walk in random environment on $ℤ^{+} = {0, 1, 2, \dots}$ , with reflection at the origin, where the random environment is subject to a vanishing perturbation. Our results complement existing criteria for random walks in random environments and for Markov chains with asymptotically zero drift, and are significantly different from the previously studied cases. Our method is based on a martingale technique—the method of Lyapunov functions. ...

The Infinite Random Order of Dimension $k$

Peter Winkler (1985)

Publications du Département de mathématiques (Lyon)

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Random differential inclusions with convex right hand sides

Krystyna Grytczuk, Emilia Rotkiewicz (1991)

Annales Polonici Mathematici

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Abstract. The main result of the present paper deals with the existence of solutions of random functional-differential inclusions of the form ẋ(t, ω) ∈ G(t, ω, x(·, ω), ẋ(·, ω)) with G taking as its values nonempty compact and convex subsets of n-dimensional Euclidean space $R^{n}$ .

Existence of graphs with sub exponential transitions probability decay and applications

Clément Rau (2010)

Bulletin de la Société Mathématique de France

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In this paper, we recall the existence of graphs with bounded valency such that the simple random walk has a return probability at time $n$ at the origin of order $exp (- n^{α}),$ for fixed $α \in [0, 1 [$ and with Følner function $exp (n^{\frac{2 α}{1 - α}})$ . This result was proved by Erschler (see [4], [3]); we give a more detailed proof of this construction in the appendix. In the second part, we give an application of the existence of such graphs. We obtain bounds of the correct order for some functional of the local time of a simple random...

Displaying similar documents to “Critical constants for recurrence of random walks on G -spaces”

Displaying similar documents to “Critical constants for recurrence of random walks on $G$ -spaces”