Title: Universality of the time constant for critical first-passage percolation on the triangular lattice
Speaker: Jack Hanson (City College, CUNY)
Abstract: We consider first-passage percolation (FPP) on the triangular lattice with vertex weights whose common distribution function $F$ satisfies $F(0) = \frac{1}{2}$. 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 $T_n$ the first-passage time from 0 to the boundary of the box of sidelength $n$, we show existence of the time constant – the limit of $T_n / \log n$ – and find its exact value to be $I / (2 (\sqrt{3\pi})$. (Here $I = \inf \{x > 0 : F(x) > \frac{1}{2} \}$.) This shows that the time constant is universal, in the sense that it is insensitive to most details of F. 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 $F$.
