WIAS Preprint No. 3023, (2023)

Weakly self-avoiding walk in a Pareto-distributed random potential


  • König, Wolfgang
    ORCID: 0000-0002-7673-4364
  • Pétrélis, Nicolas
  • Soares dos Santos, Renato
  • van Zuijlen, Willem
    ORCID: 0000-0002-2079-0359

2020 Mathematics Subject Classification

  • 60H25 60G70 82C44 60F10 60G55 60G57


  • Random walk in random potential, random variational problem, parabolic Anderson model, path localisation, intermittent islands, weakly self-avoiding walk, Poisson point process convergence, spatial extreme-value analysis, Pareto potential distribution




We investigate a model of continuous-time simple random walk paths in ℤ d undergoing two competing interactions: an attractive one towards the large values of a random potential, and a self-repellent one in the spirit of the well-known weakly self-avoiding random walk. We take the potential to be i.i.d. Pareto-distributed with parameter α > d, and we tune the strength of the interactions in such a way that they both contribute on the same scale as t → ∞. Our main results are (1) the identification of the logarithmic asymptotics of the partition function of the model in terms of a random variational formula, and, (2) the identification of the path behaviour that gives the overwhelming contribution to the partition function for α > 2d: the random-walk path follows an optimal trajectory that visits each of a finite number of random lattice sites for a positive random fraction of time. We prove a law of large numbers for this behaviour, i.e., that all other path behaviours give strictly less contribution to the partition function.The joint distribution of the variational problem and of the optimal path can be expressed in terms of a limiting Poisson point process arising by a rescaling of the random potential. The latter convergence is in distribution?and is in the spirit of a standard extreme-value setting for a rescaling of an i.i.d. potential in large boxes, like in KLMS09.

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