## View abstract

### Session S06 - Interacting Stochastic Systems

Tuesday, July 13, 12:35 ~ 13:10 UTC-3

## The time constant of finitary random interlacements

### Sarai Hernandez-Torres

#### Technion, Israel   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak15152ac5e01c27f3f7b0124350910069').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy15152ac5e01c27f3f7b0124350910069 = 's&#97;r&#97;&#105;.h' + '&#64;'; addy15152ac5e01c27f3f7b0124350910069 = addy15152ac5e01c27f3f7b0124350910069 + 'c&#97;mp&#117;s' + '&#46;' + 't&#101;chn&#105;&#111;n' + '&#46;' + '&#97;c' + '&#46;' + '&#105;l'; var addy_text15152ac5e01c27f3f7b0124350910069 = 's&#97;r&#97;&#105;.h' + '&#64;' + 'c&#97;mp&#117;s' + '&#46;' + 't&#101;chn&#105;&#111;n' + '&#46;' + '&#97;c' + '&#46;' + '&#105;l';document.getElementById('cloak15152ac5e01c27f3f7b0124350910069').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy15152ac5e01c27f3f7b0124350910069 + '\'>'+addy_text15152ac5e01c27f3f7b0124350910069+'<\/a>';

The finitary random interlacement $\text{FRI}(u, T)$ is a Poisson point process of geometrically killed random walks on $\mathbb{Z}^d$, with $d \geq 3$. The parameter $u$ modulates the intensity of the point process, while $T$ is the expected path length. Although the process lacks global monotonicity on $T$, $\text{FRI}(u, T)$ exhibits a phase transition. For $T > T^{*}(u)$, $\text{FRI}(u, T)$ defines a unique infinite connected subgraph of $\mathbb{Z}^d$ with a chemical distance. We focus on the asymptotic behavior of this chemical distance and—in particular—the time constant function. This function is a normalized limit of the chemical distance between the origin and a sequence of vertices growing in a fixed direction. In this sense, the time constant function defines an asymptotic norm. Our main result is on its continuity (as a function of the parameters of $\text{FRI}$).

Joint work with Eviatar B. Procaccia (Technion, Israel) and Ron Rosenthal (Technion, Israel).