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Critical first-passage percolation in two dimensions

Michael Damron, Georgia Tech
Monday, April 5, 2021 - 2:30pm to 3:20pm

In \$2d\$ first-passage percolation (FPP), we place nonnegative i.i.d. weights (\$t_e\$) on the edges of \$Z^2\$ and study the induced weighted graph pseudometric \$T = T(x,y)\$. If we denote by \$p = P(t_e = 0)\$, then there is a transition in the large-scale behavior of the model as \$p\$ varies from 0 to 1. When \$p < 1/2\$, \$T(0,x)\$ grows linearly in \$x\$, and when \$p > 1/2\$, it is stochastically bounded. The critical case, where \$p = 1/2\$, is more subtle, and the sublinear growth of \$T(0,x)\$ depends on the behavior of the distribution function of \$t_e\$ near zero. I will discuss my work over the past few years that (a) determines the exact rate of growth of \$T(0,x)\$, (b) determines the "time constant" for the site-FPP model on the triangular lattice and, more recently, (c) studies the growth of \$T(0,x)\$ in a dynamical version of the model, where weights are resampled according to independent exponential clocks. These are joint works with J. Hanson, D. Harper, W.-K. Lam, P. Tang, and X. Wang.

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