On the speed of convergence for two-dimensional first passage Ising percolation

Abstract

Consider first passage Ising percolation on $Z^2$. Let $\beta$ denote the reciprocal temperature and let $h$ denote an external magnetic field. Denote by $\beta_c$ the critical temperature and, for $\beta\<\beta_c$, let

h_c(\beta) = h_c = \sup\{h:\theta(\beta, h) = 0\},

where $\theta(\beta,h)$ is the probability that the origin is connected by an infinite (+)-cluster. With these definitions let us consider first passage Ising percolation on $Z^2$. Let $a_{0,n}$ denote the first passage time from $(0,0)$ to $(n,0)$. It follows from a subadditive argument that

\lim_{n \to \infty} \frac{a_{0, n}}{n} = \nu \text{ a.s. and in } L_1.

It is known that $\nu > 0$ if $\beta < \beta_c$ and $|h| < h_c(\beta)$. Here we will estimate the speed of the convergence,

\nu n \leq Ea_{0,n} \leq \nu n + C(n \log^5 n)^{1/2}

for some constant $C$. Define $\mu_{\beta,h}$ to be the unique Gibbs measure for $\beta<\beta_c$. We also prove that there exist $\tilde{C},\tilde{\alpha}>0$ such that

\mu_{\beta, h}(|a_{0,n} - Ea_{0,n} \geq x) \leq \tilde{C} \exp(\\tilde{a}\frac{x^2}{n \log^4 n})

In addition to $a_{0,n)$, we shall also discuss other passage times.