Random walk on the zero-range process

• Autor(a) principal: Weberson da Silva Arcanjo
• Data de Publicação: 2022
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Repositório Institucional da UFMG
• Texto Completo: http://hdl.handle.net/1843/50995; https://orcid.org/0000-0003-4875-3805
• topic: Random Walk; Zero-Range; Law of Large Numbers; Matemática – Teses; Passeio aleatório (Matemática) – Teses; Lei dos Grandes Números – Teses

The main theme of this thesis is the study of random walks in random environments. More specifically, we will study a random walk that moves in a medium that is inhomogeneous in $\mathbb{Z}$ and evolves with time, called dynamic random environment. In this case, the environment will be an interacting particle system in $\mathbb{Z}$ known as the zero-range process. This name comes from the fact that, in this system, particles interact with each other only when they share the same site. We will denote this process by $\eta=(\eta_{t})_{t\geq 0}$. We will assume that this process is in equilibrium. We say that site $x \in \mathbb{Z}$ is occupied by particles at time $t$ if $\eta_{t}(x)>0$ and othewise, we say that $x$ is vacant. Let $\alpha_{\bullet}, \beta_{\bullet}, \alpha_{\circ}$ and $\beta_{\circ}$ be positive real numbers. On $\eta$ we define a continuous-time, nearest-neighbor random walk $X=(X_t)_{t \geq 0}$, that starts at the origin and evolves as follows: when $X$ is on a site occupied by particles, it jumps to the right with a rate $\alpha_{\bullet}$ and to the left with a rate $\beta_{\bullet}$; when $X$ is on a vacant site, these rates are respectively $\alpha_{\circ}$ and $\beta_{\circ}$. Here, the random walk does not interfere with the evolution of the environment. We will study the asymptotic behavior of this random walk in time in the case where $X$ has a positive local drift to the right. More precisely, assuming that the differences $\alpha_{\bullet}-\beta_{\bullet}$ and $\alpha_{\circ}-\beta_{\circ}$ are sufficiently large, we will obtain a law of large numbers for $X $. That is, we will show that $X_t/t$ converges almost surely to a value $v$, which depends of the density of particles of the environment. Random walks in dynamic random environments given by interacting particle system have attracted a lot of interest in the last two decades. Several techniques have been developed and used to obtain refined results about these processes such as laws of large numbers, central limit theorems, invariance principles and large deviation estimates. Some of these techniques are regeneration, analytical techniques and multiscale renormalization. The applicability of each of the techniques and the range of results depend on the mixing properties of the environment. In this context, conservative particle systems such as the exclusion process, independent random walks and the zero-range process have drawn much attention due to the lack of good mixing properties. Thus, we hope that the results in this thesis contribute on the mathematical understanding of these processes.