Shape transition under excess self-intersections for transient random walk

Asselah, Amine

doi:10.1214/09-aihp318

Cited by 5 publications

(10 citation statements)

References 23 publications

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“…In this paper, motivated by the results of Spitzer [19] for genuinely d-dimensional random walks and the approach of Becker and König [3](see also Asselah [2] where non-integer α is also treated) we shall study the asymptotic behavior of var(L n (α)) without imposing any moment assumptions on the random walk. The central idea behind our approach is to compare the selfintersection local times L n (α) of a general d-dimensional walk with those of its symmetrised version.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The result was motivated by [19] and [3] (and improves related results of Becker and Konig for d = 3 and d = 4). Several cases treated in [2,4,5,8,10,7,3,17] can then be obtained as particular cases.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…We first give the proof for the case d = 1. As in the proof of Proposition 4 we begin from expression (2), and define the sequences p i , and δ i for i = 1, . .…”

Section: Proofs Of Main Resultsmentioning

confidence: 99%

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Optimal Bounds for the Variance of Self-Intersection Local Times

Deligiannidis

Utev

2016

International Journal of Stochastic Analysis

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For a random walk Sn, n ≥ 0 in Z d , let l(n, x) be its local time at the site x ∈ Z d . Define the α-fold self intersection local time Ln(α) := x l(n, x) α , and let Ln(α|ǫ, d) the corresponding quantity for d-dimensional simple random walk. Without imposing any moment conditions, we show that the variances of the local times var(Ln(α)) of any genuinely ddimensional random walk are bounded above by the corresponding characteristics of the simple symmetric random walk in Z d , i.e. var(Ln(α)) ≤ C var[Ln(α|ǫ, d)] ∼ K d,α v d,α (n). In particular, variances of local times of all genuinely d-dimensional random walks, d ≥ 4, are similar to the 4-dimensional symmetric case var(Ln(α)) = O(n). On the other hand, in dimensions d ≤ 3 the resemblance to the simple random walk lim infn→∞ var(Ln(α))/v d,α (n) > 0 implies that the jumps must have zero mean and finite second moment.2000 Mathematics Subject Classification. Primary 60G50, 60F05. Key words and phrases. Self-intersection local time, random walk in random scenery.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Optimal Bounds for the Variance of Self-Intersection Local Times

Deligiannidis

Utev

2016

International Journal of Stochastic Analysis

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“…The large deviations, and central limit theorem for l n q are tackled in [2]: we establish a shape transition in the walk's strategy to realize the deviations { l n q q − E[ l n q q ] ≥ nξ} with ξ > 0. This transition occurs at a critical value q c (d) = d d−2 suggesting the following picture.…”

Section: On Self-intersection Local Timesmentioning

confidence: 99%

“…Now, the aymptotic behaviour is found as we maximize βn 1/10 u − u 2 2c d , which is c d β 2 n 1/5 /2. In other words, it is a simple computation that we omit, which yields for any β > 0, lim n→∞ 1 n 1/5 log Z I (β) = c d β 2 2 .…”

Section: (52)mentioning

confidence: 99%

Annealed Lower Tails for the Energy of a Charged Polymer

Asselah¹

2009

J Stat Phys

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We consider the energy of a randomly charged polymer. We assume that only charges on the same site interact pairwise. We study the lower tails of the energy, when averaged over both randomness, in dimension three or more. As a corollary, we obtain the correct temperature-scale for the Gibbs measure.Keywords and phrases: random polymer, large deviations, random walk in random scenery, self-intersection local times.AMS 2000 subject classification numbers: 60K35, 82C22, 60J25.Running head: Lower tails for the energy a polymer.Our toy-model comes from physics, where it is used to model proteins or DNA folding. However, physicists' usual setting differs from ours by three main features. (i) Their polymer is usually quenched: a typical realization of the charges is fixed, and the average is over the walk. (ii) A short-range repulsion is included by considering random walks such as the self-avoiding walk or the directed walk. (iii) The averages are performed with respect to the the so-called Gibbs measure: a probability measure obtained from P 0 by weighting it with exp(βH n ), with real parameter β. When β is positive, the Gibbs measure favors configuration with large energy; in other words, alike charges attract each other: this models hydrophobic interactions, where the effect of avoiding the water solvent is mimicked by an attraction among hydrophobic monomers. When β is negative, alike charges repel: this models Coulomb potential, and describes also the effective repulsion between identical bases of RNA. The issue is whether there is a critical value β c (n), such that as β crosses β c (n), a phase transition occurs. For instance, Garel and Orland [14] observed a phase transition as β crosses a β c (n) ∼ 1/n, from a collapsed shape to a random-walk like shape. Kantor and Kardar [15] discussed the quenched model for the case β < 0, that is when alike charges repel. Some heuristics (dimensional analysis on the continuum version) suggests that the (upper) critical dimension is 2: for d ≥ 3, the polymer looks like a simple random walk, whereas when d < 2, its average end-to-end distance is n ν with ν = 2 d+2 . Let us also mention studies of Derrida, Griffiths and Higgs [11] and Derrida and Higgs [12]: both study the quenched Gibbs measure exp(−βH n )dP 0 , with β > 0, for a one dimensional directed random walkP 0 , and obtain evidence for a phase transition (a so-called weak freezing transition).Our interest stems from recent mathematical works of Chen [8], and Chen and Khoshnevisan [10], dealing with central limit theorems for H n . Chen [8] establishes also an annealed moderate deviation principle, under the additional assumption that E[exp(λη 2 )] < ∞, for some λ > 0. More precisely, with the annealed law denoted P , d ≥ 3, n

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