The number of open paths in an oriented $\rho$-percolation model

TL;DR

This paper investigates the exponential growth rate of open paths in an oriented $ ho$-percolation model, linking it to polymer model free energy, and provides explicit calculations and probabilistic fluctuations in certain parameter regimes.

Contribution

It establishes the asymptotic exponential growth rate of open paths, relates it to polymer free energy, and derives precise probabilistic fluctuation results in specific parameter ranges.

Findings

Number of open paths grows as $e^{n ext{alpha}( ho)}$ asymptotically.

Explicit computation of the exponent $ ext{alpha}( ho)$ in some parameter ranges.

In certain regimes, the count scales as $n^{-1/2} W e^{n ext{alpha}( ho)}$ with a nondegenerate random variable $W$.

Abstract

We study the asymptotic properties of the number of open paths of length $n$ in an oriented $\rho$-percolation model. We show that this number is $e^{n\alpha(\rho)(1+o(1))}$ as $n \to \infty$. The exponent $\alpha$ is deterministic, it can be expressed in terms of the free energy of a polymer model, and it can be explicitely computed in some range of the parameters. Moreover, in a restricted range of the parameters, we even show that the number of such paths is $n^{-1/2} W e^{n\alpha(\rho)}(1+o(1))$ for some nondegenerate random variable $W$. We build on connections with the model of directed polymers in random environment, and we use techniques and results developed in this context.