Challenges in Probability and Mathematical Physics

Abstract

In this thesis we focus on three problems in probability and mathematical physics. The first problem is the emergence of infinite cycles in the interchange process. In the interchange process on a graph $G=(V,E)$, distinguished particles are placed on the vertices of $G$ with independent Poisson clocks on the edges. When the clock of an edge rings, the two particles on the two sides of the edge interchange. In this way, a random permutation $\pi _\beta:V\to V$ is formed for any time $\beta >0$. One of the main objects of study is the cycle structure of the random permutation and the emergence of long cycles. We prove the existence of infinite cycles in the interchange process on $\mathbb Z ^d$ for all dimensions $d\ge 5$ and all large $\beta $, establishing a conjecture of Bálint Tóth from 1993 in these dimensions. The second problem is the asymptotic behavior of the one-dimensional multi-particle Diffusion Limited Aggregation (MDLA) at its critical density $\lambda=1$. Previous works have verified that the size of the aggregate $X_t$ at time $t$ is $t^{1/2}$ in the subcritical regime and linear in the supercritical regime. This paper establishes the conjecture that the growth rate at criticiality is $t^{2/3}$. Moreover, we derive the scaling limit proving that $$\{ t^{-2/3}X_{st} \}_{s\ge 0} \overset{\textnormal{d}}{\to} \Big\{ \int_0^s Z_u du \Big\}_{s\ge 0}, \quad t\to \infty , $$ where the speed process $\{Z_t\}$ is a $(-1/3)$-self-similar diffusion given by $Z_t = (3V_t)^{-2/3}$, where~$V_t$ is the $\frac{8}{3}$-Bessel process. The third problem is the phenomenon of coalescence of geodesics in first-passage percolation. First passage percolation is the study of the random metric obtain by independently sampling the weight of each edge in a graph. The main object of study is the shortest paths (geodesics) in this metric and their lengths. We prove that in $\mathbb Z ^2$ geodesics with nearby starting and ending points have significant overlap, coalescing on all but small portions near their endpoints. The statement is quantified, with power-law dependence of the involved quantities on the length of the geodesics.