Open Access
February 2017 Variational formulas and disorder regimes of random walks in random potentials
Firas Rassoul-Agha, Timo Seppäläinen, Atilla Yilmaz
Bernoulli 23(1): 405-431 (February 2017). DOI: 10.3150/15-BEJ747


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.


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Firas Rassoul-Agha. Timo Seppäläinen. Atilla Yilmaz. "Variational formulas and disorder regimes of random walks in random potentials." Bernoulli 23 (1) 405 - 431, February 2017.


Received: 1 October 2014; Revised: 1 April 2015; Published: February 2017
First available in Project Euclid: 27 September 2016

zbMATH: 1368.60105
MathSciNet: MR3556777
Digital Object Identifier: 10.3150/15-BEJ747

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

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability

Vol.23 • No. 1 • February 2017
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