Preprint C177/2014
Metastability of the ABC model in the zero-temperature limit and of reversible random walks in potential fields

Ricardo Misturini

**Keywords: **
Metastability | Scaling limits | ABC Model | Brownian motion | Reversible random walks | Exit points

In this thesis we study the phenomenon of metastability in two specific contexts.

In the first part of the text we consider the zero-temperature limit of the ABC Model. The ABC model is a conservative stochastic dynamics consisting of three species of particles, labeled $A$, $B$, $C$, on a discrete ring $\{-N,\ldots,N\}$ (one particle per site). The system evolves by nearest neighbor transpositions: $AB \to BA$, $BC \to CB$, $CA \to AC$ with rate $q$ and $BA\to AB$, $CB\to BC$, $AC\to CA$ with rate $1$. We investigate a strongly asymmetric regime, the zero-temperature limit, where $q=e^{-\beta}$, $\beta\uparrow\infty$. The main result asserts that the particles almost always form three pure domains (one of each species) and that, as the system size $N$ grows with $\beta$, this segregated shape evolves (in a proper time-scale) as a Brownian motion on the circle, which may have a drift.

In the second part we consider reversible random walks in potential fields. More precisely, let $\Xi$ be an open and bounded subset of $\mathbb R^d$ and let $F:\Xi \to \mathbb R$ be a twice continuously differential function. Denote by $\Xi_N$ the discretization of $\Xi$, $\Xi_N=\Xi\cap(N^{-1}\mathbb Z^d)$, and denote by $\{X_N(t): t\geq0\}$ the continuous-time, nearest-neighbor, random walk on $\Xi_N$ which jumps from $x$ to $y$ at rate $e^{-(1/2)N[F(y) - F(x)]}$. We examine the metastable behavior of $\{X_N(t): t\geq0\}$ among the wells of the potential $F$. Our main result states that, in an appropriate time-scale, the evolution of the random walk on $\Xi_N$ can be described by a random walk in a weighted graph, in which the vertices represent the wells of the force field and the edges represent the saddle points.