Final Report Summary - RIMMP (Random and Integrable Models in Mathematical Phyiscs)
Random matrix models play an important role in various intimately related problems, such as
(i) The distribution of the eigenvalues of random matrices is given by matrix integrals. These studies have their origin in the study of energy levels of heavy nuclei (Wigner, Dyson, Mehta). Universality of the distribution arising in the large size limit of the matrix integrals is one of the main issues. It is closely related to the integrability of the differential equations governing these distributions; the last decade has seen an explosion of results in that area.
(ii) Non-intersecting random walks and Brownian motions.
(iii) Integrals over groups and symmetric spaces lead to a variety of interesting matrix models, which satisfy non-linear integrable differential equations.
The main goal of the project has been to put the theory of random matrices, random permutations and random walks in the context of the theory of integrable systems. This goal has been achieved by a significant progress in the theory of integrable systems and their perturbations, including its analytic, geometric and numerical aspects. Perturbative classification of integrable partial differential equations (PDEs) and their discrete analogues, various approaches to analytically constructing their exact solutions, the analysis of singular limits of important classes of solutions, the spectral geometry of Riemann surfaces, new asymptotic methods involving Painlevé transcendents and theta-functions, efficient numerical approaches to solving dispersive PDEs, all these techniques have been further developed. The strong connection of these problems with asymptotic problems in random matrices has helped to a deeper understanding of both fields of research.
The moduli space of integrable hierarchies of topological type with n dependent variables coincides with the moduli space of semisimple Frobenius manifolds. The latter coincides with the space of monodromy data of certain Riemann - Hilbert problem for a n×n matrix linear differential operator with rational coefficients. More recently integrable hierarchies of a wider class called deformed integrable hierarchies proved to be important in the framework of topological field theories. These deformed integrable hierarchies called the Hodge hierarchies, possess a Hamiltonian structure, and a tau-function for each of its solution. In the particular case of quantum cohomology, the tau-function of a solution to the hierarchy generates the intersection numbers of the Gromov--Witten classes and their descendants along with the characteristic classes of Hodge bundles on the moduli spaces of stable maps. For the one-dimensional Frobenius manifold the Hodge hierarchy is a deformation of the Korteweg--de Vries (KdV) hierarchy depending on an infinite number of parameters. Conjecturally this hierarchy is a universal object in the class of scalar Hamiltonian integrable hierarchies possessing tau-functions.
A significant work has been done in the study of critical behavior of solutions to Hamiltonian perturbations of hyperbolic and elliptic systems. In particular, the universal nature of this critical behavior is remarkable. For hyperbolic one- and two-component systems as well as for elliptic two-component systems, it has been conjectured that solutions can be asymptotically described in a certain critical regime by special solutions to the Painlevé I equation and its hierarchy. Such conjecture has been tested via numerical comparison by obtaining the numerical solution of the Painlevé transcendents relevant to the problem. Such Painlevé transcendents appeared a decade earlier in the literature in the description of critical regimes in matrix models. Several numerical techniques have been developed to study nonlinear dispersive local and nonlocal PDE near blow up and critical regimes. Such investigations have pointed out the extreme similarities of behaviour of solutions of integrable and non integrable equations showing that critical regimes have generically an integrable structure. The quantitative study of dispersive shock waves in two spatial dimensions has been numerically challenging. Such study has shown that such behaviour has many similiarities with the one-dimensional case.
The main reason to look for an integrable structure in random matrices, random processes and related problems is the possibility to solve the problem in close form. Usually a determinantal structure is underlying integrability. The study of the asymptotic behaviour of the model is then obtained by applying a very powerful technique called steepest descent method for oscillatory Riemann-Hilbert problems. Such technique was developed in the seminal work of P. Deift and X. Zhou on the long time asymptotic of a nonlinear integrable dispersive equations and now it has a wide range of applications in random matrices, classical theory of Toeplitz and Hankel determinants and models of statistical mechanics.
The main results concerning random matrices are the following.
The random matrix model with external source was studied very succesfully. A general even potential has been considered A vector equilibrium problem was used, and very surprisingly it was discovered that the Pearcey phase transition is not always generic but can cross a Painlevé phase transition. The two matrix model was analyzed for special potentials which give rise to a new critical phenomenon. In particular it turns out that the two-point correlator kernels can be described by a Riemann-Hilbert problem that was first obtained in a completely different model for two groups of non-intersecting Brownian paths that touch each other at the tacnode point.
An unexpected development which was not foreseen in the original proposal is the use of Riemann-Hilbert techniques in the analysis of normal matrix models. The analysis could be done for the normal matrix model with various potential with symmetries. Another unexpected development of the proposal is due to the surprising discovery by Akemann and Burda that eigenvalues or singular values of products of Ginibre matrices have a determinantal structure. This structure has enabled the asymptotic study of a large set of new matrix models. Explicit formulas for the correlation kernels which allowed to describe scaling limits near the hard edge have been found. It results in a new family of limiting kernels that probably have a universal character.
In the framework of classical theory of Toeplitz and Hankel determinants the proof of the long-stranding conjecture of Basor and Tracy concerning the asymptotics of the Toeplitz determinant with the Fisher-Hartwig weight of the most general form has been obtained. Similar asymptotic formula for Hankel and Hankel + Toeplitz determinants have been obtained as well.
The six-vertex model is one of the basic two-dimensional models of statistical mechanics. The celebrated Izergin-Korepin formula gives an exact solution for the partition function of the six-vertex model with domain wall boundary conditions (DWBC). Such partition function has been studied in the large n limit for several phase regions of the phase diagram.
Many of the asymptotic regimes studied above involved Painlevé transcendents. One of the most striking discovery of Painlevé equations of the last decades is its relation to conformal field theory. Namely the tau-function of solutions of Painlevé equation is related to four-point Virasoro conformal blocks with central charge c=1. This connection allows to evaluate the constant pre-factors in the asymptotics of the tau-functions corresponding to the general solutions of Painlevé equations. This is a very important new development in the theory of Painlevé equations. Indeed, this type of constants have been previously evaluated only for very special types of solutions. At the same time, the pre-factors in the asymptotics of tau-functions, starting from the classical works of Onsager on Ising model, have been the principle challenge in the theory of exactly solvable statistical models as they carry the principal physical features of the models.
(i) The distribution of the eigenvalues of random matrices is given by matrix integrals. These studies have their origin in the study of energy levels of heavy nuclei (Wigner, Dyson, Mehta). Universality of the distribution arising in the large size limit of the matrix integrals is one of the main issues. It is closely related to the integrability of the differential equations governing these distributions; the last decade has seen an explosion of results in that area.
(ii) Non-intersecting random walks and Brownian motions.
(iii) Integrals over groups and symmetric spaces lead to a variety of interesting matrix models, which satisfy non-linear integrable differential equations.
The main goal of the project has been to put the theory of random matrices, random permutations and random walks in the context of the theory of integrable systems. This goal has been achieved by a significant progress in the theory of integrable systems and their perturbations, including its analytic, geometric and numerical aspects. Perturbative classification of integrable partial differential equations (PDEs) and their discrete analogues, various approaches to analytically constructing their exact solutions, the analysis of singular limits of important classes of solutions, the spectral geometry of Riemann surfaces, new asymptotic methods involving Painlevé transcendents and theta-functions, efficient numerical approaches to solving dispersive PDEs, all these techniques have been further developed. The strong connection of these problems with asymptotic problems in random matrices has helped to a deeper understanding of both fields of research.
The moduli space of integrable hierarchies of topological type with n dependent variables coincides with the moduli space of semisimple Frobenius manifolds. The latter coincides with the space of monodromy data of certain Riemann - Hilbert problem for a n×n matrix linear differential operator with rational coefficients. More recently integrable hierarchies of a wider class called deformed integrable hierarchies proved to be important in the framework of topological field theories. These deformed integrable hierarchies called the Hodge hierarchies, possess a Hamiltonian structure, and a tau-function for each of its solution. In the particular case of quantum cohomology, the tau-function of a solution to the hierarchy generates the intersection numbers of the Gromov--Witten classes and their descendants along with the characteristic classes of Hodge bundles on the moduli spaces of stable maps. For the one-dimensional Frobenius manifold the Hodge hierarchy is a deformation of the Korteweg--de Vries (KdV) hierarchy depending on an infinite number of parameters. Conjecturally this hierarchy is a universal object in the class of scalar Hamiltonian integrable hierarchies possessing tau-functions.
A significant work has been done in the study of critical behavior of solutions to Hamiltonian perturbations of hyperbolic and elliptic systems. In particular, the universal nature of this critical behavior is remarkable. For hyperbolic one- and two-component systems as well as for elliptic two-component systems, it has been conjectured that solutions can be asymptotically described in a certain critical regime by special solutions to the Painlevé I equation and its hierarchy. Such conjecture has been tested via numerical comparison by obtaining the numerical solution of the Painlevé transcendents relevant to the problem. Such Painlevé transcendents appeared a decade earlier in the literature in the description of critical regimes in matrix models. Several numerical techniques have been developed to study nonlinear dispersive local and nonlocal PDE near blow up and critical regimes. Such investigations have pointed out the extreme similarities of behaviour of solutions of integrable and non integrable equations showing that critical regimes have generically an integrable structure. The quantitative study of dispersive shock waves in two spatial dimensions has been numerically challenging. Such study has shown that such behaviour has many similiarities with the one-dimensional case.
The main reason to look for an integrable structure in random matrices, random processes and related problems is the possibility to solve the problem in close form. Usually a determinantal structure is underlying integrability. The study of the asymptotic behaviour of the model is then obtained by applying a very powerful technique called steepest descent method for oscillatory Riemann-Hilbert problems. Such technique was developed in the seminal work of P. Deift and X. Zhou on the long time asymptotic of a nonlinear integrable dispersive equations and now it has a wide range of applications in random matrices, classical theory of Toeplitz and Hankel determinants and models of statistical mechanics.
The main results concerning random matrices are the following.
The random matrix model with external source was studied very succesfully. A general even potential has been considered A vector equilibrium problem was used, and very surprisingly it was discovered that the Pearcey phase transition is not always generic but can cross a Painlevé phase transition. The two matrix model was analyzed for special potentials which give rise to a new critical phenomenon. In particular it turns out that the two-point correlator kernels can be described by a Riemann-Hilbert problem that was first obtained in a completely different model for two groups of non-intersecting Brownian paths that touch each other at the tacnode point.
An unexpected development which was not foreseen in the original proposal is the use of Riemann-Hilbert techniques in the analysis of normal matrix models. The analysis could be done for the normal matrix model with various potential with symmetries. Another unexpected development of the proposal is due to the surprising discovery by Akemann and Burda that eigenvalues or singular values of products of Ginibre matrices have a determinantal structure. This structure has enabled the asymptotic study of a large set of new matrix models. Explicit formulas for the correlation kernels which allowed to describe scaling limits near the hard edge have been found. It results in a new family of limiting kernels that probably have a universal character.
In the framework of classical theory of Toeplitz and Hankel determinants the proof of the long-stranding conjecture of Basor and Tracy concerning the asymptotics of the Toeplitz determinant with the Fisher-Hartwig weight of the most general form has been obtained. Similar asymptotic formula for Hankel and Hankel + Toeplitz determinants have been obtained as well.
The six-vertex model is one of the basic two-dimensional models of statistical mechanics. The celebrated Izergin-Korepin formula gives an exact solution for the partition function of the six-vertex model with domain wall boundary conditions (DWBC). Such partition function has been studied in the large n limit for several phase regions of the phase diagram.
Many of the asymptotic regimes studied above involved Painlevé transcendents. One of the most striking discovery of Painlevé equations of the last decades is its relation to conformal field theory. Namely the tau-function of solutions of Painlevé equation is related to four-point Virasoro conformal blocks with central charge c=1. This connection allows to evaluate the constant pre-factors in the asymptotics of the tau-functions corresponding to the general solutions of Painlevé equations. This is a very important new development in the theory of Painlevé equations. Indeed, this type of constants have been previously evaluated only for very special types of solutions. At the same time, the pre-factors in the asymptotics of tau-functions, starting from the classical works of Onsager on Ising model, have been the principle challenge in the theory of exactly solvable statistical models as they carry the principal physical features of the models.