A large-deviation result for the range of random walk and for the Wiener sausage

“…Upward large deviations are obtained by Hamana and Kesten [HK01]. Their result implies that there exists a positive function J d , such that for ε ∈ (0, 1 − κ d ),…”

Section: Preliminariesmentioning

confidence: 91%

On the nature of the Swiss cheese in dimension $3$

Asselah¹,

Schapira²

2020

Ann. Probab.

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We study scenarii linked with the Swiss cheese picture in dimension three obtained when two random walks are forced to meet often, or when one random walk is forced to squeeze its range. In the case of two random walks, we show that they most likely meet in a region of optimal density. In the case of one random walk, we show that a small range is reached by a strategy uniform in time. Both results rely on an original inequality estimating the cost of visiting sparse sites, and in the case of one random walk on the precise Large Deviation Principle of van den Berg, Bolthausen and den Hollander [BBH01], including their sharp estimates of the rate functions in the neighborhood of the origin.Remarkably the Swiss cheese heuristic also highlight a crucial difference between dimension 3 and dimensions 5 and higher. Indeed, in dimension three the typical scenario is time homogeneous, in the sense that the Wiener sausage considered up to time t, would spend all its time localized in a region of typical scale (t/ε) 1/3 , filling a fraction of order ε of every volume element, when the

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“…Assuming that we can expand 1 m log E[e −uQm ] as u ↓ 0 (we will also take m 1/u), we get . When trying to optimise over m, we realise that we need to take u 2 m 3/2 m −1/2 (and the term u log m will turn out to be negligible): taking m = cu −1 (where the constant c is chosen so as to optimise the parenthesis above), we get that [6] that the range R n of simple random walk satisfies an upward large deviation principle for d ≥ 2. Namely, they showed that the limit The following conjecture deals with the upward large deviations of the range trimmed when the local times exceed a certain threshold.…”

Section: Appendix a Bridge Estimatesmentioning

confidence: 99%

Annealed scaling for a charged polymer in dimensions two and higher

Berger¹,

Hollander²,

Poisat³

2018

J. Phys. A: Math. Theor.

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This paper considers an undirected polymer chain on Z d , d ≥ 2, with i.i.d. random charges attached to its constituent monomers. Each self-intersection of the polymer chain contributes an energy to the interaction Hamiltonian that is equal to the product of the charges of the two monomers that meet. The joint probability distribution for the polymer chain and the charges is given by the Gibbs distribution associated with the interaction Hamiltonian. The object of interest is the annealed free energy per monomer in the limit as the length n of the polymer chain tends to infinity.We show that there is a critical curve in the parameter plane spanned by the charge bias and the inverse temperature separating an extended phase from a collapsed phase. We derive the scaling of the critical curve for small and for large charge bias and the scaling of the annealed free energy for small inverse temperature. We show that in a subset of the collapsed phase the polymer chain is subdiffusive, namely, on scale (n/ log n) 1/(d+2) it moves like a Brownian motion conditioned to stay inside a ball with a deterministic radius and a randomly shifted center. We expect this scaling to hold throughout the collapsed phase. We further expect that in the extended phase the polymer chain scales like a weakly self-avoiding walk.The scaling of the critical curve for small charge bias and the scaling of the annealed free energy for small inverse temperature are both anomalous. Proofs are based on a detailed analysis for simple random walk of the downward large deviations of the selfintersection local time and the upward large deviations of the range. Part of our scaling results are rough. We formulate conjectures under which they can be sharpened. The existence of the free energy remains an open problem, which we are able to settle in a subset of the collapsed phase for a subclass of charge distributions.

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A large-deviation result for the range of random walk and for the Wiener sausage

Cited by 29 publications

References 16 publications

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

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

On the nature of the Swiss cheese in dimension $3$

Annealed scaling for a charged polymer in dimensions two and higher

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