We consider a one dimensional random walk in a random environment (RWRE) with a positive speed limn→∞ (Xn/) = υα > 0. Gantert and Zeitouni showed that if the environment has both positive and negative local drifts then the quenched slowdown probabilities P ω(Xn < xn) with x∈ (0,υα) decay approximately like exp{- n1-1/s} for a deterministic s > 1. More precisely, they showed that n -γ log Pω(Xn < xn) converges to 0 or -∞ depending on whether γ > 1 - 1/s or γ < 1 - 1/ s. In this paper, we improve on this by showing that n -1+1/s log P ω(Xn< xn) oscillates between 0 and -∞ , almost surely.
","abstract_html":"<p>We consider a one dimensional random walk in a random environment (RWRE) with a positive speed lim<em>n</em>→∞ (<em>Xn/</em>) = υα > 0. Gantert and Zeitouni showed that if the environment has both positive and negative local drifts then the quenched slowdown probabilities <em>P</em> ω(<em>Xn</em> < <em>xn</em>) with <em>x</em>∈ (0,υα) decay approximately like exp{-<em> n</em>1-1/<em>s</em>} for a deterministic <em>s</em> > 1. More precisely, they showed that <em>n</em> -γ log <em>P</em>ω(<em>Xn </em>< <em>xn</em>) converges to 0 or -∞ depending on whether γ > 1 - 1/<em>s</em> or γ < 1 - 1/<em> s</em>. In this paper, we improve on this by showing that <em>n</em> -1+1/<em>s</em> log <em>P</em> ω(Xn< <em>xn</em>) oscillates between 0 and -∞ , almost surely.</p>","abstract_has_math":false,"creators":["Ahn, Sung Won"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Jonathon Peterson","Rodrigo Banuelos","Mark Daniel Ward","Nung Kwan Yip"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016-08-01T07:00:00Z","date_published":"2016-08-01T07:00:00Z","updated_at":"2026-07-24T03:53:55Z","subjects":["Pure sciences","Large deviation","Probability","Random walk in random environment","Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://docs.lib.purdue.edu/open_access_dissertations/734","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Jonathon Peterson","Rodrigo Banuelos","Mark Daniel Ward","Nung Kwan Yip"]},{"key":"dc:creator","label":"Author","values":["Ahn, Sung Won"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Pure sciences","Large deviation","Probability","Random walk in random environment","Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://docs.lib.purdue.edu/open_access_dissertations/734"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We consider a one dimensional random walk in a random environment (RWRE) with a positive speed limn→∞ (Xn/) = υα > 0. Gantert and Zeitouni showed that if the environment has both positive and negative local drifts then the quenched slowdown probabilities P ω(Xn < xn) with x∈ (0,υα) decay approximately like exp{- n1-1/s} for a deterministic s > 1. More precisely, they showed that n -γ log Pω(Xn < xn) converges to 0 or -∞ depending on whether γ > 1 - 1/s or γ < 1 - 1/ s. In this paper, we improve on this by showing that n -1+1/s log P ω(Xn< xn) oscillates between 0 and -∞ , almost surely.
"]},{"key":"dc:title","label":"Title","values":["Oscillation of quenched slowdown asymptotics of random walks in random environment in Z"]}]}],"canonical_facts":{"dc:contributor":["Jonathon Peterson","Rodrigo Banuelos","Mark Daniel Ward","Nung Kwan Yip"],"dc:creator":["Ahn, Sung Won"],"dc:description.abstract":["We consider a one dimensional random walk in a random environment (RWRE) with a positive speed limn→∞ (Xn/) = υα > 0. Gantert and Zeitouni showed that if the environment has both positive and negative local drifts then the quenched slowdown probabilities P ω(Xn < xn) with x∈ (0,υα) decay approximately like exp{- n1-1/s} for a deterministic s > 1. More precisely, they showed that n -γ log Pω(Xn < xn) converges to 0 or -∞ depending on whether γ > 1 - 1/s or γ < 1 - 1/ s. In this paper, we improve on this by showing that n -1+1/s log P ω(Xn< xn) oscillates between 0 and -∞ , almost surely.
"],"dc:identifier":["https://docs.lib.purdue.edu/open_access_dissertations/734"],"dc:subject":["Pure sciences","Large deviation","Probability","Random walk in random environment","Mathematics"],"dc:title":["Oscillation of quenched slowdown asymptotics of random walks in random environment in Z"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T03:53:55Z"}