Acknowledgements

1. Prof. Sofendi, Ph,D., Universitas Negeri Sriwijaya, Palembang2. Dr. Rita Inderawati, M.Pd., Universitas Negeri Sriwijaya, Palembang3. Prof. Dr. Indawan Syahri, M.Pd., Universitas Muhammadiyah, Palembang4. Dr. Sunda Ariana, M.Pd., M.M., Universitas Bina Darma, Palembang5. Dr. Dian Ekawati, M.Pd., Lembaga Penjamin Mutu Pendidikan, Palembang6. Dr. A. Gumawang Jati, M.A., Institut Teknologi Bandung, Bandung7. Dr. Yuli Utanto, M.Si., Universitas Negeri Semarang, Semarang8. Dr. Een Y. Haenilah, Universitas Negeri Lampung, Lampung9. Dr. Nurlaelah, M.Hum., Universitas Muslim Indonesia, Makassar10. Drs. Deddy Suryana, M.A., Universitas Pendidikan Indonesia, Bandung11. Dr. Alimin Laundung, M.Pd., Politeknik Negeri Makassar, Makassar12. Drs. Zubaidi, Dipl. TESL, M.Pd., Politeknik Negeri Malang, Malang13. I Nyoman Rajin Aryana, S.Pd., M.Hum., Politeknik Negeri Bali, Bali14. Dr. Mulyadi, M.A., Universitas PGRI, Palembang15. Drs. Iman Suroso, M.Pd., Politeknik Negeri Semarang, SemarangTh

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Large deviations for random walk in a random environment

In this work, we study the large deviation properties of random walk in a random environment on $\mathbb{Z}^d$ with $d\geq1$. We start with the quenched case, take the point of view of the particle, and prove the large deviation principle (LDP) for the pair empirical measure of the environment Markov chain. By an appropriate contraction, we deduce the quenched LDP for the mean velocity of the particle and obtain a variational formula for the corresponding rate function $I_q$. We propose an Ansatz for the minimizer of this formula. This Ansatz is easily verified when $d=1$. In his 2003 paper, Varadhan proves the averaged LDP for the mean velocity and gives a variational formula for the corresponding rate function $I_a$. Under the non-nestling assumption (resp. Kalikow's condition), we show that $I_a$ is strictly convex and analytic on a non-empty open set $\mathcal{A}$, and that the true velocity $\xi_o$ is an element (resp. in the closure) of $\mathcal{A}$. We then identify the minimizer of Varadhan's variational formula at any $\xi\in\mathcal{A}$. For walks in high dimension, we believe that $I_a$ and $I_q$ agree on a set with non-empty interior. We prove this for space-time walks when the dimension is at least 3+1. In the latter case, we show that the cheapest way to condition the asymptotic mean velocity of the particle to be equal to any $\xi$ close to $\xi_o$ is to tilt the transition kernel of the environment Markov chain via a Doob $h$-transform.Comment: 82 pages. PhD thesis. Advisor: S.R.S. Varadha

Multitype Contact Process on Z\Z: Extinction and Interface

We consider a two-type contact process on $\Z$ in which both types have equal finite range and supercritical infection rate. We show that a given type becomes extinct with probability 1 if and only if, in the initial configuration, it is confined to a finite interval $[-L,L]$ and the other type occupies infinitely many sites both in $(-\infty, L)$ and $(L, \infty)$. We also show that, starting from the configuration in which all sites in $(-\infty, 0]$ are occupied by type 1 particles and all sites in $(0, \infty)$ are occupied by type 2 particles, the process $\rho_t$ defined by the size of the interface area between the two types at time $t$ is tight