Speaker: Michael Damron (Georgia Institute of Technology)

 

Title: Percolation and first-passage percolation on logarithmic subsets of \(Z²\)

 

 

Abstract:  Abstract: In two-dimensional Bernoulli percolation, we declare each edge of the square grid \(Z²\) to be open with probability \(p\) or closed with probability \(1 − p\), independently from edge to edge. There is a critical value \(p_c = 1/2\), such that for \(p < p_c\) all components of open edges are finite, and for \(p > p_c\) there is a unique infinite component of open edges. In 1983, Grimmett introduced the following variant. Let \(f\) be a nonnegative real function on \([0, ∞)\), and consider the subgraph \(G_f of Z²\) induced by the edges between the positive first coordinate axis and the graph of \(f\). Grimmett found that if \(f(u) ~ a log u\) as \(u → ∞\), the critical value \(p_c(f)\) for percolation on \(G_f\) equals a specific function of \(a\). In 1986, Chayes–Chayes considered the function \(f(u) = a log(1 + u) + b log(1 + log(1 + u))\) and showed that if \(b > 2a\), then the percolation \(G_f\) has an infinite open component at the critical point (i.e., a discontinuous phase transition). In joint work with Wai-Kit Lam, we prove that the phase transition is discontinuous if and only if \(b > a\), and we compute sharp asymptotics for all \(p\), \(a\), and \(b\) of the expected passage time in \(G_f\) from the origin to the vertical line \(x = n\) in the related first-passage percolation model, improving results of Ahlberg. We also find asymptotics for the variance and a central limit theorem.

 

For more details, please visit https://webhome.auburn.edu/~lzc0090/SASA/20251001_Damron.html .

 

Host: Le Chen