An expansion for self-interacting random walks

doi:10.1214/10-BJPS121

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An expansion for self-interacting random walks

Journal Article•DOI•

An expansion for self-interacting random walks

Remco van der Hofstad¹, Mark Holmes¹•Institutions (1)

01 Feb 2012-Brazilian Journal of Probability and Statistics (Brazilian Statistical Association)-Vol. 26, Iss: 1, pp 1-55

TL;DR: In this article, the authors derive a perturbation expansion for general self-interacting random walks, where steps are made on the basis of the history of the path, and show that the expansion gives rise to useful formulae for the speed and variance of the random walk, when these quantities are known to exist.

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Abstract: We derive a perturbation expansion for general self-interacting random walks, where steps are made on the basis of the history of the path. Examples of models where this expansion applies are reinforced random walk, excited random walk, the true (weakly) self-avoiding walk, loop-erased random walk, and annealed random walk in random environment. In this paper we show that the expansion gives rise to useful formulae for the speed and variance of the random walk, when these quantities are known to exist. The results and formulae of this paper have been used elsewhere by the authors to prove monotonicity properties for the speed (in high dimensions) of excited random walk and related models, and certain models of random walk in random environment. We also derive a law of large numbers and central limit theorem (with explicit error terms) directly from this expansion, under strong assumptions on the expansion coefficients. The assumptions are shown to be satisfied by excited random walk in high dimensions with small excitation parameter, a model of reinforced random walk with underlying drift and small reinforcement parameter, and certain models of random walk in random environment under strong ellipticity conditions.

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Excited random walks: results, methods, open problems.

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Elena Kosygina¹, Martin P. W. Zerner²•Institutions (2)

City University of New York¹, University of Tübingen²

01 Jan 2013

TL;DR: A survey of known results and some of the methods and several new results for transient one-dimensional excited random walks in bounded i.i.d. cookie environments can be found in this paper.

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Abstract: We consider a class of self-interacting random walks in deterministic or random environments, known as excited random walks or cookie walks, on the d-dimensional integer lattice. The main purpose of this paper is two-fold: to give a survey of known results and some of the methods and to present several new results. The latter include functional limit theorems for transient one-dimensional excited random walks in bounded i.i.d. cookie environments as well as some zero-one laws. Several open problems are stated. 1. Model Description Random walks (RWs) and their scaling limits are probably the most widely known and frequently used stochastic processes in probability theory, mathematical physics, and applications. Studies of a RW in a random medium are an attempt to understand which macroscopic effects can be seen and modeled by subjecting the RW’s dynamics on a microscopic level to various kinds of noise, for example, allowing it to interact with a random Received March 30, 2012. AMS Subject Classification: 60K35, 60K37, 60J80.

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54 citations

Journal Article•DOI•

Monotonicity for excited random walk in high dimensions

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Remco van der Hofstad¹, Mark Holmes²•Institutions (2)

Eindhoven University of Technology¹, University of Auckland²

01 May 2010-Probability Theory and Related Fields

TL;DR: In this article, it was shown that the drift θ(d, β) for excited random walk in dimension d is monotone in the excitement parameter when d ≥ 9.

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Abstract: We prove that the drift θ(d, β) for excited random walk in dimension d is monotone in the excitement parameter ${\beta \in [0,1]}$ , when d is sufficiently large. We give an explicit criterion for monotonicity involving random walk Green’s functions, and use rigorous numerical upper bounds provided by Hara (Private communication, 2007) to verify the criterion for d ≥ 9.

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30 citations

Posted Content•

An expansion for self-interacting random walks

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Remco van der Hofstad¹, Mark Holmes¹•Institutions (1)

Eindhoven University of Technology¹

05 Jun 2007-arXiv: Probability

TL;DR: In this article, the authors derived a perturbation expansion for general self-interacting random walks, where steps are made on the basis of the history of the path, and showed that the expansion gives rise to useful formulae for the speed and variance of the random walk, when these quantities are known to exist.

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25 citations

Journal Article•DOI•

A monotonicity property for random walk in a partially random environment

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Mark Holmes¹, Rongfeng Sun²•Institutions (2)

University of Auckland¹, National University of Singapore²

01 Apr 2012-Stochastic Processes and their Applications

TL;DR: In this paper, a law of large numbers for random walks in certain kinds of i.i.d. random environments in Z d was proved, which is an extension of a result of Bolthausen et al. (2003) [4].

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14 citations

Cites background or methods from "An expansion for self-interacting r..."

...However in this example the velocity is indeed monotone in p. Holmes and Salisbury [12] prove that this is the case for any 2-valued environment....
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...1 of [8], shows that the series in the speed formula (1....
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...For any translation invariant self-interacting random walk (see [8] for precise details) for which Eo[Xn −Xn−1] converges, the formula (1....
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...This analysis would require different estimates, similar to those used in the analysis of once-reinforced random walk with drift in [8]....
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...The lace expansion for self-interacting random walks of van der Hofstad and Holmes [8] gives the following series representation for the expected increment of the RWRE under Po, Eo[Xn −Xn−1] = Eo[X1] + n∑ m=2 ∑ x xπm(x), (1.5) where πm(x), for m ≥ 2, x ∈ Zd are somewhat complicated quantities known as lace-expansion coefficients....
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Posted Content•

The continuous-time lace expansion

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David C. Brydges, Tyler Helmuth¹, Mark Holmes²•Institutions (2)

Durham University¹, University of Melbourne²

23 May 2019-arXiv: Probability

TL;DR: In this paper, the authors derive a continuous-time lace expansion for a broad class of self-interacting continuous time random walks and apply it to the lattice Edwards model at weak coupling.

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Abstract: We derive a continuous-time lace expansion for a broad class of self-interacting continuous-time random walks. Our expansion applies when the self-interaction is a sufficiently nice function of the local time of a continuous-time random walk. As a special case we obtain a continuous-time lace expansion for a class of spin systems that admit continuous-time random walk representations. We apply our lace expansion to the $n$-component $g|\varphi|^4$ model on $\mathbb{Z}^{d}$ when $n=1,2$, and prove that the critical Green's function $G_{ u_{c}}(x)$ is asymptotically a multiple of $|x|^{2-d}$ when $d\geq 5$ at weak coupling. As another application of our method we establish the analogous result for the lattice Edwards model at weak coupling.

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9 citations

Cites background from "An expansion for self-interacting r..."

...The lace expansion has since been reformulated in many different settings: unoriented and oriented percolation [23,37,55], the contact process [54], lattice trees and animals [24], Ising and gj'j(4) models [39, 40], the random connection model [28], and various self-interacting random walk models [21, 27, 50, 52]....
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References

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Book•

Large Deviations Techniques and Applications

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Amir Dembo, Ofer Zeitouni

27 Mar 1998

TL;DR: The LDP for Abstract Empirical Measures and applications-The Finite Dimensional Case and Applications of Empirically Measures LDP are presented.

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Abstract: LDP for Finite Dimensional Spaces.- Applications-The Finite Dimensional Case.- General Principles.- Sample Path Large Deviations.- The LDP for Abstract Empirical Measures.- Applications of Empirical Measures LDP.

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5,578 citations

"An expansion for self-interacting r..." refers background in this paper

...(6.21) Thus, by Cramér’s theorem (e.g., see Dembo and Zeitouni (1998), Theorem 2.2.30) there exists J = J (D(·),w0(·)) > 0 such that Q0(ωn = ω0) ≤ e−Jn for all n....
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Journal Article•DOI•

A survey of random processes with reinforcement

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Robin Pemantle¹•Institutions (1)

University of Pennsylvania¹

12 Feb 2007-Probability Surveys

TL;DR: The models surveyed in this paper include generalized Polya urns, reinforced random walks, interacting urn models, and continuous reinforced processes, with a focus on methods and results, with sketches provided of some proofs.

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Abstract: The models surveyed include generalized Polya urns, reinforced random walks, interacting urn models, and continuous reinforced processes. Emphasis is on methods and results, with sketches provided of some proofs. Applications are discussed in statistics, biology, economics and a number of other areas.

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617 citations

"An expansion for self-interacting r..." refers background or methods in this paper

...See Pemantle (2007) for a survey of self-interacting random walks with reinforcement....
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...of of this fact is currently missing. Similarly, it has not been proved that the speed for once-reinforced random walk on the tree is monotone decreasing in the reinforcement parameter (see [9]). See [28] for a survey of self-interacting random walks with reinforcement. In the past decades, the lace expansion has proved to be an extremely useful technique to investigate a variety of models above their...
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Journal Article•DOI•

Mean-Field Critical Behaviour for Percolation in High Dimensions

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Takashi Hara¹, Gordon Slade²•Institutions (2)

Courant Institute of Mathematical Sciences¹, McMaster University²

01 Mar 1990-Communications in Mathematical Physics

TL;DR: In this article, the authors used an expansion related to the lace expansion for a self-avoiding walk to prove that the triangle condition is satisfied for independent bond percolation on the d-dimensional hypercubic lattice, ifd is sufficiently large.

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Abstract: The triangle condition for percolation states that$\sum\limits_{x,y} {\tau (0,x)\tau (0,y) \cdot \tau (y,0)} $ is finite at the critical point, where τ(x, y) is the probability that the sitesx andy are connected. We use an expansion related to the lace expansion for a self-avoiding walk to prove that the triangle condition is satisfied in two situations: (i) for nearest-neighbour independent bond percolation on thed-dimensional hypercubic lattice, ifd is sufficiently large, and (ii) in more than six dimensions for a class of “spread-out” models of independent bond percolation which are believed to be in the same universality class as the nearest-neighbour model. The class of models in (ii) includes the case where the bond occupation probability is constant for bonds of length less than some large number, and is zero otherwise. In the course of the proof an infrared bound is obtained. The triangle condition is known to imply that various critical exponents take their mean-field (Bethe lattice) values$(\gamma = \beta = 1,\delta = \Delta _t = 2, t\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 2)$ and that the percolation density is continuous at the critical point. We also prove thatv2 in (i) and (ii), wherev2 is the critical exponent for the correlation length.

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234 citations

Journal Article•DOI•

Self-Avoiding Walk in 5 or More Dimensions

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David C. Brydges¹, Thomas Spencer²•Institutions (2)

University of Virginia¹, Courant Institute of Mathematical Sciences²

01 Mar 1985-Communications in Mathematical Physics

TL;DR: In this paper, it was shown that for a T step weakly self-avoiding random walk in five or more dimensions the variance of the endpoint is of orderT and the scaling limit is gaussian, asT→∞.

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Abstract: Using an expansion based on the renormalization group philosophy we prove that for aT step weakly self-avoiding random walk in five or more dimensions the variance of the endpoint is of orderT and the scaling limit is gaussian, asT→∞.

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226 citations

"An expansion for self-interacting r..." refers background in this paper

...Examples are self-avoiding walks above four dimensions (Brydges and Spencer, 1985; Hara and Slade, 1992; Slade 1987, 1988, 1989), lattice trees above eight dimensions (Derbez and Slade 1997, 1998; Hara and Slade, 1990b; Holmes, 2008), the contact process above four dimensions (van der Hofstad and…...
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...Examples are self-avoiding walks above four dimensions (Brydges and Spencer, 1985; Hara and Slade, 1992; Slade 1987, 1988, 1989), lattice trees above eight dimensions (Derbez and Slade 1997, 1998; Hara and Slade, 1990b; Holmes, 2008), the contact process above four dimensions (van der Hofstad and Sakai 2004, 2010), oriented percolation above four dimensions (van der Hofstad and Slade, 2003; Nguyen and Yang 1993, 1996), and percolation above six dimensions (Hara and Slade 1990a, 2000a, 2000b)....
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Journal Article•DOI•

Phase transition in reinforced random walk and RWRE on trees

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Robin Pemantle

01 Jul 1988-Annals of Probability

TL;DR: In this paper, the authors show that the reinforced random walk can vary from transient to recurrent, depending on the value of an adjustable parameter measuring the strength of the feedback, which is calculated at the phase transition.

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Abstract: A random walk on an infinite tree is given a particular kind of positive feedback so edges already traversed are more likely to be traversed in the future. Using exchangeability theory, the process is shown to be equivalent to a random walk in a random environment (RWRE), that is to say, a mixture of Markov chains. Criteria are given to determine whether a RWRE is transient or recurrent. These criteria apply to show that the reinforced random walk can vary from transient to recurrent, depending on the value of an adjustable parameter measuring the strength of the feedback. The value of the parameter at the phase transition is calculated.

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189 citations

"An expansion for self-interacting r..." refers background in this paper

...A few examples are self-reinforced random walks (Durrett, Kesten and Limic, 2002; Pemantle, 1988; Rolles, 2002), excited random walks (Benjamini and Wilson, 2003; Kozma 2003, 2005; Zerner 2005, 2006), true-self avoiding walks and loop-erased random walks....
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...A few examples are self-reinforced random walks [8, 24, 25], excited random walks [2, 20, 21, 29, 30], true-self avoiding walks and loop-erased random walks....
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