Loop Condensation Effects in the Behavior of Random Walks

Khanin, Konstantin; Mazel, A.; Shlosman, Senya; Sinaĭ, Ya. G.

doi:10.1007/978-1-4612-0279-0_9

Cited by 17 publications

(30 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In their influential paper [KM94], Khanin, Mazel, Shlosman and Sinai make the surprising observation that even in high dimensions the random variables counting the total number of intersections of two independent simple random walks have subexponential tails at infinity. They argue that this observation is intimately related to the fact that a random walk in random potential has subdiffusive behaviour in all dimensions.…”

Section: Motivationmentioning

confidence: 99%

See 1 more Smart Citation

Upper tails for intersection local times of random walks in supercritical dimensions

Chen¹,

Mörters

2008

Journal of the London Mathematical Society

View full text Add to dashboard Cite

show abstract

Section: Motivationmentioning

confidence: 99%

“…we are assuming that we are in supercritical dimensions. Then, in the case of simple, symmetric random walks, Khanin et al show in [KM94] that there exist constants c 1 , c 2 > 0 such that, for all a large enough, exp − c 1 a…”

Section: Motivationmentioning

confidence: 99%

Upper tails for intersection local times of random walks in supercritical dimensions

Chen¹,

Mörters

2008

Journal of the London Mathematical Society

View full text Add to dashboard Cite

show abstract

“…Motivation. In a paper that appeared in "The 1994 Dynkin Festschrift", Khanin, Mazel, Shlosman and Sinai [9] considered the following problem. Let S(n), n ∈ N 0 , be the simple random walk on Z d and let R = {z ∈ Z d : S(n) = z for some n ∈ N 0 } (1.1) be its infinite-time range.…”

Section: Introduction and Main Results: Theorems 1-6mentioning

confidence: 99%

“…In Section 1.6 we briefly look at the intersection volume of three or more Wiener sausages. In Section 1.7 we discuss the discrete space-time setting considered in [9]. In Section 1.8 we give the outline of the rest of the paper.…”

Section: Wiener Sausages Let β(T) Tmentioning

confidence: 99%

On the volume of the intersection of two Wiener sausages

Berg

Bolthausen

Hollander³

2004

Ann. Math.

View full text Add to dashboard Cite

For a > 0, let W a 1 (t) and W a 2 (t) be the a-neighbourhoods of two independent standard Brownian motions in R d starting at 0 and observed until time t. We prove that, for d ≥ 3 and c > 0,and derive a variational representation for the rate constant I κa d (c). Here, κ a is the Newtonian capacity of the ball with radius a. We show that the optimal strategy to realise the above large deviation is for W a 1 (ct) and W a 2 (ct) to "form a Swiss cheese": the two Wiener sausages cover part of the space, leaving random holes whose sizes are of order 1 and whose density varies on scale t 1/d according to a certain optimal profile.We study in detail the function c → I We also derive the analogous result for d = 2, namely, 742M. VAN DEN BERG, E. BOLTHAUSEN, AND F. DEN HOLLANDER where the rate constant has the same variational representation as in d ≥ 3 after κ a is replaced by 2π. In this case I 2π 2 (c) = Θ 2 (2πc)/2π with Θ 2 (u) < ∞ if and only if u ∈ (u , ∞) and Θ 2 is strictly decreasing on (u , ∞) with a zero limit.Acknowledgment.

show abstract

“…Intersections of random paths have been an extensively studied topic, not only due to its fundamental importance in the theory of stochastic processes, but also because they play a central role in various physical models as the "polaron problem" in quantum field theory [18], the parabolic Anderson model describing diffusion in random potential ( [24,32,8]), or random polymer models ( [17,19,20]). Therefore, various mathematical objects have been introduced to quantify these intersections: the range of a random walk, the volume of the Wiener sausage, or the self-intersection local times, which is the main object of the present paper.…”

Section: Introductionmentioning

confidence: 99%