Analytical Methods and Field Theory for Disordered Systems

Abstract

This thesis presents several aspects of the physics of disordered elastic systems and of the analytical methods used for their study. On one hand we will be interested in universal properties of avalanche processes in the statics and dynamics (at the depinning transition) of elastic interfaces of arbitrary dimension in disordered media at zero temperature. To study these questions we will use the functional renormalization group. After a review of these aspects we will more particularly present the results obtained during the thesis on (i) the spatial structure of avalanches and (ii) the correlations between avalanches. On the other hand we will be interested in static properties of directed polymers in $1+1$ dimension, and in particular in observables related to the KPZ universality class. In this context the study of exactly solvable models has recently led to important progress. After a review of these aspects we will be more particularly interested in exactly solvable models of directed polymer on the square lattice and present the results obtained during the thesis in this direction: (i) classification of Bethe ansatz exactly solvable models of directed polymer at finite temperature on the square lattice; (ii) KPZ universality for the Log-Gamma and Inverse-Beta models; (iii) KPZ universality and non-universality for the Beta model; (iv) stationary measures of the Inverse-Beta model and of related zero temperature models.

PhD Thesis. 172+18pages

References in corpus (22)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Burgers Turbulence
• Growing interfaces uncover universal fluctuations behind scale invariance
• Scaling description of the yielding transition in soft amorphous solids at zero temperature
• Asymptotics in ASEP with Step Initial Condition
• An exact solution for the KPZ equation with flat initial conditions
• The one-dimensional KPZ equation and its universality class
• Crackling dynamics in material failure as the signature of a self-organized dynamic phase transition
• A Fredholm Determinant Representation in ASEP
• Exactly solvable model of avalanches dynamics for Barkhausen crackling noise
• The KPZ equation with flat initial condition and the directed polymer with one free end
• Creep motion of an elastic string in a random potential
• Local dynamics of a randomly pinned crack front during creep and forced propagation: An experimental study
• Numerical Calculation of the Functional renormalization group fixed-point functions at the depinning transition
• An exact mapping of the stochastic field theory for Manna sandpiles to interfaces in random media
• The One-dimensional KPZ Equation and the Airy Process
• The strict-weak lattice polymer
• Statics and dynamics of elastic manifolds in media with long-range correlated disorder
• Short time growth of a KPZ interface with flat initial conditions
• Aftershock production rate of driven viscoelastic interfaces
• The universal high temperature regime of pinned elastic objects
• Universality for directed polymers in thin rectangles