Contest based on a directed polymer in a random medium

TL;DR

This paper introduces a two-player game inspired by a directed polymer model, revealing a traveling wave distribution of scores and a phase transition where one player can achieve maximum score, with insights into wave velocity selection.

Contribution

It presents a novel game model based on directed polymers, analyzing its score distribution and phase transition, and connects the front velocity to nonlinear stability mechanisms.

Findings

Score distribution follows a traveling wave form with double exponential decay.

A phase transition occurs allowing one player to reach maximum score.

The front velocity is determined by nonlinear marginal stability mechanisms.

Abstract

We introduce a simple one-parameter game derived from a model describing the properties of a directed polymer in a random medium. At his turn, each of the two players picks a move among two alternatives in order to maximize his final score, and minimize opponent's return. For a game of length $n$, we find that the probability distribution of the final score $S_n$ develops a traveling wave form, ${\rm Prob}(S_n=m)=f(m-v n)$, with the wave profile $f(z)$ unusually decaying as a double exponential for large positive and negative $z$. In addition, as the only parameter in the game is varied, we find a transition where one player is able to get his maximum theoretical score. By extending this model, we suggest that the front velocity $v$ is selected by the nonlinear marginal stability mechanism arising in some traveling wave problems for which the profile decays exponentially, and for which standard traveling wave theory applies.