2019/04/26 by Michael Damron, Damron, Michael, Jack Hanson +3
Mathematics · #Stochastic processes and statistical mechanics #Random Matrices and Applications #Markov Chains and Monte Carlo Methods
paper · pdf · doi:10.48550/arxiv.1904.12009
We consider first-passage percolation (FPP) on the triangular lattice with vertex weights (tv) whose common distribution function F satisfies F(0)=1/2. This is known as the critical case of FPP because large (critical) zero-weight clusters allow travel between distant points in time which is sublinear in the distance. Denoting by T(0,∂ B(n)) the first-passage time from 0 to \x : ‖x‖_∞ = n\, we show existence of the "time constant'' and find its exact value to be limn → ∞ (T(0,∂ B(n)))/(log n) = (I)/(2√(3)π) almost surely, where I = inf\x > 0 : F(x) > 1/2\ and F is any critical distribution for tv. This result shows that the time constant is universal and depends only on the value of I. Furthermore, we find the exact value of the limiting normalized variance, which is also only a function of I, under the optimal moment condition on F. The proof method also shows an analogous universality on other two-dimensional lattices, assuming the time constant exists.