[Paper Review] Analytical Methods and Field Theory for Disordered Systems

Thimothée Thiery|arXiv (Cornell University)|May 21, 2017

Random Matrices and Applications200 references3 citations

TL;DR

This thesis develops analytical field-theory methods to study disordered elastic systems, focusing on avalanche dynamics at the depinning transition and exact solvability in directed polymers. Using functional renormalization group (FRG) and Bethe ansatz techniques, it establishes universal scaling laws and identifies new exactly solvable models in the KPZ universality class, including Log-Gamma and Inverse-Beta models, with explicit stationary measures and non-universal behavior in the Beta model at finite temperature.

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.

Motivation & Objective

• To understand universal scaling properties of avalanche processes in elastic interfaces at zero temperature and at the depinning transition.
• To analyze the spatial structure and correlations of avalanches in disordered elastic systems using functional renormalization group (FRG) techniques.
• To classify exactly solvable models of directed polymers on the square lattice at finite temperature.
• To establish KPZ universality for specific models such as Log-Gamma and Inverse-Beta polymers.
• To investigate the interplay between exact solvability and universality in the Beta model, revealing both universal and non-universal behavior.

Proposed method

• Application of the functional renormalization group (FRG) to study statics and dynamics of disordered elastic interfaces at zero temperature.
• Use of FRG to analyze avalanche dynamics at the depinning transition, including correlations and spatial structure.
• Classification of Bethe ansatz-integrable models of directed polymers on the square lattice with short-range elasticity.
• Exact solution of the continuum directed polymer model using algebraic integrability and stationary measure techniques.
• Derivation of universal scaling limits and finite-size corrections via exact solvability and Fredholm determinant methods.
• Analysis of non-universal behavior in the Beta model by comparing finite-temperature results with KPZ fixed-point predictions.

Experimental results

Research questions

• RQ1 What are the universal scaling laws governing the spatial structure and correlations of avalanches in disordered elastic interfaces?
• RQ2 How do functional renormalization group methods describe the depinning transition and avalanche dynamics in elastic systems?
• RQ3 Which directed polymer models on the square lattice are exactly solvable via the Bethe ansatz at finite temperature?
• RQ4 To what extent do the Log-Gamma and Inverse-Beta models exhibit KPZ universality in their scaling limits?
• RQ5 What is the nature of the stationary measure in the Inverse-Beta model, and how does it relate to zero-temperature models and KPZ universality?

Key findings

• The FRG approach successfully captures the universal scaling of avalanche size and spatial correlations in elastic interfaces at the depinning transition.
• The spatial structure of avalanches is found to be self-similar with a universal fractal dimension, consistent with theoretical predictions.
• The Log-Gamma and Inverse-Beta models are identified as exactly solvable via the Bethe ansatz, enabling exact computation of the free energy and correlation functions.
• The KPZ universality class is confirmed for the Log-Gamma and Inverse-Beta models through exact scaling limits matching the KPZ fixed point.
• The Beta model exhibits both universal and non-universal behavior: while the tail of the free energy distribution matches the Tracy-Widom GUE distribution, finite-size corrections deviate from universality.
• The stationary measure of the Inverse-Beta model is derived exactly, providing a bridge between finite-temperature models and zero-temperature interface models.