Some integrable models in the KPZ universality class

Abstract : This thesis is about exactly solvable models in the KPZ universality class. The first chapter provides an overview of the recent methods designed to study such systems. We also present the different works which constitute this thesis, leaving aside the technical details, but rather focusing on the interpretation of the results and the general methods that we use. The three next chapters each correspond to an article published or submitted for publication. The first chapter is an asymptotic study of the q-TASEP interacting particle system, when the system is perturbed by a few slower particles. We show that the system obeys the same limit theorem as TASEP, and one observes the so-called BBP transition. The second chapter, based on a work in collaboration with Ivan Corwin, introduces new exactly solvable exclusion processes. We verify the predictions from KPZ scaling theory, and we also study the less universal behaviour of the first particle. The third chapter corresponds to a second work in collaboration with Ivan Corwin. We introduce a random walk in random environment, which turns out to be exactly solvable. We prove that the second order correction to the large deviation principle is Tracy-Widom distributed on a cube root scale. We give a probabilistic interpretation of this limit theorem, and show that the result also propagates at zero-temperature.
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Submitted on : Wednesday, June 24, 2015 - 7:01:48 PM
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  • HAL Id : tel-01167855, version 1

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Guillaume Barraquand. Some integrable models in the KPZ universality class. Probability [math.PR]. Université Paris Diderot -- Paris 7, 2015. English. ⟨tel-01167855⟩

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