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Self-intersection local times of random walks: exponential moments in subcritical dimensions
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  • Published: 23 June 2011

Self-intersection local times of random walks: exponential moments in subcritical dimensions

  • Mathias Becker1 &
  • Wolfgang König1,2 

Probability Theory and Related Fields volume 154, pages 585–605 (2012)Cite this article

Abstract

Fix p > 1, not necessarily integer, with p(d − 2) < d. We study the p-fold self-intersection local time of a simple random walk on the lattice \({\mathbb Z^d}\) up to time t. This is the p-norm of the vector of the walker’s local times, ℓ t . We derive precise logarithmic asymptotics of the expectation of exp{θ t ||ℓ t || p } for scales θ t > 0 that are bounded from above, possibly tending to zero. The speed is identified in terms of mixed powers of t and θ t , and the precise rate is characterized in terms of a variational formula, which is in close connection to the Gagliardo–Nirenberg inequality. As a corollary, we obtain a large-deviation principle for ||ℓ t || p /(tr t ) for deviation functions r t satisfying \({t r_t\gg \mathbb E[||\ell_t||_p]}\) . Informally, it turns out that the random walk homogeneously squeezes in a t-dependent box with diameter of order ≪ t 1/d to produce the required amount of self-intersections. Our main tool is an upper bound for the joint density of the local times of the walk.

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Authors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117, Berlin, Germany

    Mathias Becker & Wolfgang König

  2. Technical University Berlin, Str. des 17. Juni 136, 10623, Berlin, Germany

    Wolfgang König

Authors
  1. Mathias Becker
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  2. Wolfgang König
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Correspondence to Wolfgang König.

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Becker, M., König, W. Self-intersection local times of random walks: exponential moments in subcritical dimensions. Probab. Theory Relat. Fields 154, 585–605 (2012). https://doi.org/10.1007/s00440-011-0377-0

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  • Received: 23 July 2010

  • Revised: 18 January 2011

  • Published: 23 June 2011

  • Issue Date: December 2012

  • DOI: https://doi.org/10.1007/s00440-011-0377-0

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Keywords

  • Self-intersection local time
  • Upper tail
  • Donsker–Varadhan large deviations
  • Variational formula
  • Gagliardo–Nirenberg inequality

Mathematics Subject Classification (2000)

  • 60K37
  • 60F10
  • 60J55
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