Variational Formulas and Cocycle solutions for Directed Polymer and Percolation Models

Abstract

We discuss variational formulas for the law of large numbers limits of certain models of motion in a random medium: namely, the limiting time constant for last-passage percolation and the limiting free energy for directed polymers. The results are valid for models in arbitrary dimension, steps of the admissible paths can be general, the environment process is ergodic under spatial translations, and the potential accumulated along a path can depend on the environment and the next step of the path. The variational formulas come in two types: one minimizes over gradient-like cocycles, and another one maximizes over invariant measures on the space of environments and paths. Minimizing cocycles can be obtained from Busemann functions when these can be proved to exist. The results are illustrated through 1+1 dimensional exactly solvable examples, periodic examples, and polymers in weak disorder.F. Rassoul-Agha and N.

Variational Formula for the Time Constant of First‐Passage Percolation

“…It is interesting to note that quite coincidentally, our results and those of Georgiou et al [16] [23] published their viscous stochastic homogenization result. In contrast to Lions and Souganidis [27], their proof technique has the flavor of a duality principle and has a minimax theorem at its core.…”

“…At zero-temperature, this too can be seen as an optimalcontrol problem. However, as mentioned earlier, variational formulas for directed last-passage percolation and zero-temperature polymer models have been proved in considerable generality by Georgiou et al [16].…”

“…This work was considerably generalized by Rassoul-Agha et al [33] and Rassoul-Agha and Seppäläinen [32]. Subsequently, Georgiou et al [16] extended these ideas to prove variational formulas for the directed random polymer, and for last-passage percolation.…”

“…It will be interesting to explore the behavior of the so-called integrable models under the variational formula. This has already been begun in the context of last-passage percolation and polymer models by Georgiou et al [16]. …”

“…Once we posted our preprint [24] on the arXiv, the concurrent but independent work of Georgiou et al [16] appeared. They prove discrete variational formulas for the directed polymer model at zero and finite temperature, and for the closely related last-passage percolation model.…”