Dissertation

2017

#### Degree Name

Doctor of Philosophy

Mathematics

#### Major

Applied Mathematics

Arka P. Ghosh

Alexander Roitershtein

#### Abstract

The topic of this thesis is random walks in a sparse random environment (RWSRE) on \$\mathbb{Z}\$. Basic asymptotic properties of this model were investigated by Matzavinos, Roitershtein and Seol (2016). The purpose of this work is to prove large deviation principles accompanying laws of large numbers for the position of the particle and first hitting times, which have been establish in previous work.

Large deviation principles (LDP) for random walks in i.i.d. environments were first obtained by Greven and den Hollander (1994). Using a different approach, the LDP's were extended to ergodic environments by Comets, Gantert and Zeitouni (1998, 2000). Several refitments of this result due to the same group of authors, Peres, Pisztora, and Povel have appeared since then. An alternative method of studying the large deviations for random walks in random environments (RWRE) was subsequently suggested by Vardhan and further developed in the work of Yilmaz, Rassoul-Agha, and Rosenbluth.

In this work we obtain quenched and annealed LDP for the RWSRE using a relation between the underlying RWSRE and a random walk in a dual stationary environment, which was introduced by Matzavinos, Roitershtein, and Seol. We first investigate a relation between the sparse environment and its stationary dual, and then obtain LDP's for a random walk in the stationary (and ergodic) dual environment. Next, we transform the quenched LDP in the dual setting to obtain a quenched LDP for the corresponding RWSRE and give a description of the rate function. Finally, we show that the annealed LDP in the dual setting is directly related to an annealed LDP for the RWSRE when the lengths of the cycles are bounded. Our study of the rate functions relies on the approach of Comets, Dembo, Gantert and Zeitouni (2000, 2004).

#### DOI

https://doi.org/10.31274/etd-180810-4915

Kubilay Dagtoros

en

application/pdf

53 pages

COinS