### Non-equilibrium Dynamics and Random Matrices

During the academic year 2013-2014, the School of Mathematics conducted a special program on Non-equilibrium Dynamics and Random Matrices. This program was led by Distinguished Visiting Professor Horng-Tzer Yau from Harvard University, and Thomas Spencer (IAS). Yau is a leading expert on both random matrix theory and non-equilibrium dynamics. He was joined by senior participants L. Erdös, J. Fröhlich, J. Imbrie, J. Lebowitz, J. Quastel, and H. Spohn who spent the year at IAS.

**Background and Motivation:**

The study of large random matrices in physics originated in the 1950s from the work of E. Wigner, who used them to predict the energy level statistics of a large nucleus. Today, these (Wigner-Dyson) statistics seem to describe a wide class of strongly correlated systems and they are expected to obey a universal law which depends only on the symmetry class. They appear in many parts of physics and mathematics, including statistics, number theory, quantum chaos, combinatorics, communication theory, and quantum dynamics in a random environment. In number theory, extensive numerical studies show that the zeros of the Riemann zeta function exhibit Wigner-Dyson statistics. Proving the observed universality for a broad class of systems was one of the central themes pursued this year, and it will remain an active domain of research for many years to come.

Non-equilibrium dynamics describe systems which are in flux. Examples include fluid dynamics, weather, and models in the financial and biological sciences. Although non-equilibrium dynamics are ubiquitous in the real world, they constitute one of the most challenging domains of mathematics and physics. Unfortunately, we still have only limited mathematical tools to analyze them.

This year’s program focused on two particular models of stochastic dynamics: the Kardar-Parisi-Zhang (KPZ) equation and Dyson Brownian motion. It is a remarkable fact that fluctuations of the KPZ solutions are very closely connected to random matrix theory. The KPZ equation is a stochastic equation in one dimension which has elements of “stochastic integrability.” It is also related to asymptotic representation theory. Dyson Brownian motion describes the dynamics of eigenvalues of random matrices. The work of Erdös, Schlein, Yau, and Yin proved that these dynamics reach local equilibrium very rapidly. Their result resolved some longstanding conjectures about the universality of eigenvalue spacing.

**The program had four main themes:**

- The foundations of the universality of random matrices. The program aims to unlock the deep, underlying reasons as to why random matrix statistics appear to be ubiquitous. More specifically, we would like to understand the spectral statistics for non-mean field type models, such as band matrices or the random Schrödinger equation.
- KPZ universality.
- Applications of random matrix theory to other areas of mathematics and to large data statistics, combinatorics, and condensed matter physics.
- Topics in non-equilibrium statistical physics.

In the approach initiated by L. Erdös, B. Schlein, H.T. Yau, and J. Yin, the basic conceptual reason for the emergence of universality is the fast relaxation of the Dyson Brownian motion to local equilibrium. To this end, a method combining Green’s function estimate, entropy decay, and regularity theory of parabolic differential equations was developed. It was very effective for Wigner matrices and other mean-field type matrices (such as Erdös-Renyi graphs). In order to apply this idea to non-mean field type matrices, such as band matrices or the random Schrödinger equation, several participants of the program (including the aforementioned names, as well as P. Bourgade) have gradually pursued the idea of using the quantum unique ergodicity (QUE) as a tool for going beyond mean-field theory. The QUE was a concept pioneered by Rudnick and Sarnak in their work on manifolds. It plays an important role here because, heuristically, a non-mean field model effectively behaves like a mean-field one, provided that QUE holds for this model. During this year, concrete progress was made towards proving QUE for random matrices, and its application to non-mean field models is very promising.

Progress was also made on universality at fixed energy. Due to a lack of an algebraic formula, the fixed energy universality had remained an open problem for the real symmetric matrices. However, by using a pointwise homogenization idea to interpolate between microscopic and mesoscopic scales, this last main problem in the class of the Wigner-Dyson-Mehta conjecture was resolved.

Although universality at the scale of eigenvalue spacing has been the main focus of many universality results, interesting questions emerge regarding the mesoscale fluctuÃ‚Âations. It turns out that the subleading terms to the well-known Altshuler-Shklovskii formula for the band matrix were not completely accurate. This was demonstrated in a joint work by Knowles and Erdös.

A substantial part of the program centered around the KPZ equation. Originally, the equation was proposed by Kardar, Parisi, and Zhang (1986) to describe the motion of growing fronts. Large scale mathematical activities started in the seminal work of Deift and Johansson (2000). The wide interest in the KPZ equation stems from its role in connecting seemingly different mathematical worlds, in particular, Dyson’s Brownian motion, quantum Toda chain and related integrable models, statistical mechanics of line ensembles, directed polymers in a random medium, tilings, stochastic lattice gases, and stochastic conservation laws in one dimension. During the program, several introductory lectures were given on the KPZ equation and the recent progress in that direction. The speakers included I. Borodin, J. Quastel, H. Spohn, I. Corwin, and H. Sasamoto.

During the program, a wide spectrum of advances were achieved. We list only a few examples:

- Universality of the KPZ equation. Based on heuristic arguments, one expects the KPZ equation to be “universal” in the sense that microscopic models of surface growth presenting either weak noise or a weak asymmetry is expected to converge to the KPZ equation when viewed at a suitable large scale. Some results were recently established rigorously for a class of continuous models. (M. Hairer, J. Quastel).
- Statistical fluctuations of the Toom interface. At low noise level, the Toom automation has two non-equilibrium stationary states. Through Dobrushin type boundary conditions, one can force their spatial coexistence. It is established that the resulting interface fluctuations belong to the KPZ universality class with flat initial conditions resulting in GOE edge statistics and Airy1 process. (P. Ferrari, J. Lebowitz, H. Spohn).
- Multi-component KPZ equations. One-dimensional classical and quantum fluids have generically three conservation laws: mass, momentum, and energy. Their fluctuating hydrodynamics maintains nonlinearities up to second order and extends the KPZ equation to several components. Predictions based on this approach are confimed by extensive numerical simulations. For some special cases, proofs are available. (Bernardin, Huse, Kulkarni, Olla, Spohn).

Another major objective of this year’s special program has been to understand the applications of random matrix theory. We have had lectures on condensed matter physics (D. Huse, Y. Fyodorov, P. Wiegmann), biology (S. Lieber), engineering (by A. Singer), number theory (J. Keating, P. Sarnak), quantum information theory (E. Lieb, E. Carlen), and statistics (H. Liu). These lectures provide excellent maps for future directions in random matrix theory. Interesting progress has also been made toward applications of current methods of statistics. The result (by Knowles-Yau-Yin), roughly speaking, provides a very precise description regarding when the principal component analysis is correct. In condensed matter physics, J. Imbrie recently established impressive mathematical results about localization for strongly disordered quantum interacting spin systems. Localization for interacting systems has been the subject of a great deal of activity within theoretical physics. The precise mathematical problem was proposed by David Huse at Princeton University (Physics) and a frequent participant in the program.

**Additional progress in Non-equilibrium physics.**

- J. Fröhlich addressed problems about the foundations of quantum mechanics and the effective dynamics of quantum particles interacting with the environment such as a quantized wave medium. A striking phenomenon encountered in the study of quantum particles interacting with a suitable environment is the emergence of particle tracks, as originally analyzed by Sir Nevill Mott but understood more precisely only very recently.
- J. Lebowitz and his collaborators constructed a large class of Lypunov functionals to study the behavior of classically driven sytems. He also continued his research on the foundations the Fourier law. His recent paper studied a chain of interacting Hamiltonian systems perturbed by noise. The Green-Kubo formula for the heat conduction was analyzed as a function of the coupling and noise.

**Workshops:**

In order to survey the current status of random matrix theory and non-equilibrium statistical physics, two workshops were organized in this program. The first one, Workshop on Non-equilibrium Dynamics and Random Matrices, was held in November 4-8, 2013; the second one, Workshop on Random Matrices and Random Systems, was held in March 31-April 4, 2014. The first workshop was organized by Yan Fyodorov (University of London), Herbert Spohn (TUM/IAS) and Horng-Tzer Yau (Harvard/IAS) and the second one by Gerard Ben Arous (Courant Inst.), Kurt Johansson (KTH) and Horng-Tzer Yau (Harvard/IAS). The main topics addressed in these workshops are the universality of the random matrix theory, the KPZ equation, non-equilibrium dynamics, quantum information theory, random Schrödinger operators, and the applications of random matrix theory to physics, statistics, combinatorics, and engineering. The first workshop was aimed to provide a broad survey of the current theories and included many more speculative lectures regarding various applications of random matrix theory; the second one emphasizes more mathematical rigorous and definite works. These two workshops together provided a good picture of the current status and also possible future directions.