Bernoulli

  • Bernoulli
  • Volume 23, Number 1 (2017), 405-431.

Variational formulas and disorder regimes of random walks in random potentials

Firas Rassoul-Agha, Timo Seppäläinen, and Atilla Yilmaz

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Abstract

We give two variational formulas (qVar1) and (qVar2) for the quenched free energy of a random walk in random potential (RWRP) when (i) the underlying walk is directed or undirected, (ii) the environment is stationary and ergodic, and (iii) the potential is allowed to depend on the next step of the walk which covers random walk in random environment (RWRE). In the directed i.i.d. case, we also give two variational formulas (aVar1) and (aVar2) for the annealed free energy of RWRP. These four formulas are the same except that they involve infima over different sets, and the first two are modified versions of a previously known variational formula (qVar0) for which we provide a short alternative proof. Then, we show that (qVar0) always has a minimizer, (aVar2) never has any minimizers unless the RWRP is an RWRE, and (aVar1) has a minimizer if and only if the RWRP is in the weak disorder regime. In the latter case, the minimizer of (aVar1) is unique and it is also the unique minimizer of (qVar1), but (qVar2) has no minimizers except for RWRE. In the case of strong disorder, we give a sufficient condition for the nonexistence of minimizers of (qVar1) and (qVar2) which is satisfied for the log-gamma directed polymer with a sufficiently small parameter. We end with a conjecture which implies that (qVar1) and (qVar2) have no minimizers under very strong disorder.

Article information

Source
Bernoulli, Volume 23, Number 1 (2017), 405-431.

Dates
Received: October 2014
Revised: April 2015
First available in Project Euclid: 27 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.bj/1475001359

Digital Object Identifier
doi:10.3150/15-BEJ747

Mathematical Reviews number (MathSciNet)
MR3556777

Zentralblatt MATH identifier
1368.60105

Keywords
directed polymer KPZ universality large deviation quenched free energy random environment random potential random walk strong disorder variational formula very strong disorder weak disorder

Citation

Rassoul-Agha, Firas; Seppäläinen, Timo; Yilmaz, Atilla. Variational formulas and disorder regimes of random walks in random potentials. Bernoulli 23 (2017), no. 1, 405--431. doi:10.3150/15-BEJ747. https://projecteuclid.org/euclid.bj/1475001359


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