## Final Report Summary - RHP-RMT (The Riemann-Hilbert Problem and Random Matrix Theory)

Project context and objectives

Regarding the project's objectives, 1 and 2, the fellow has computed the formula for Toeplitz determinants with Fisher-Hartwig singularities using the Borodin-Okounkov formula. This is an interesting idea to deal with a problem that was previously solved via several different methods, such as the use of orthogonal polynomials and factorisations of Toeplitz matrices. Fisher-Hartwig symbols first appeared in connection with problems in statistical mechanics. More recently they have appeared in many diverse physical applications in random matrix theory, string theory and combinatorics. Our ongoing work, which started during the last phase of the project, is concerned with the asymptotics of Toeplitz determinants with transition-type singularities and their applications in random matrix theory using the powerful Riemann-Hilbert method and the Painleve transcendents. Part of the motivation comes from the connection to entanglement entropy and, in particular, we are dealing with questions related to entanglement in quantum spin chains and, as an application, the two-dimensional Ising model. Certain quantum phase transitions which occur in interacting lattice systems at zero temperature are of particular interest. Under these conditions the system is in the ground state (which is also a pure state), and any correlations must be a consequence of the fact that the ground state is entangled. Regarding expected concrete results, we deal with gliding singularities, and aim to obtain uniform asymptotics of Toeplitz determinants as singularities coalesce. In the future, we wish to study even more complicated (matrix-valued) singularities and consider their applications in random matrix theory and integrable systems; the ultimate goal is to obtain a complete classification of transition type singularities with uniform asymptotics of Toeplitz determinants with coalescing singularities.

It is worth noting that the asymptotics of orthogonal polynomials, random matrix theory and Riemann-Hilbert problems are very closely related to the properties of Toeplitz operators, and in particular the fellow has written a few articles in this direction, which somewhat deviates from Annex I but is quite closely connected to Objectives 1 and 2. A long standing problem is to find a complete characterisation of bounded (or compact) Toeplitz operators acting on certain function spaces, such as Bergman and Fock spaces. This difficult problem is present even in the Hilbert space setting. We have made good progress on these fundamental properties of Toeplitz operators generated by distributional symbols. In particular we showed that the membership of the symbols in a certain weighted Sobolev space is sufficient for the Toeplitz operator to be bounded, and showed the natural relation of the hyperbolic geometry of the disk and the order of the distribution. Similar results for compactness were also derived. It is worth noting that our results provided a large class of bounded and compact Bergman-space Toeplitz operators, whose symbols are far beyond bounded and even locally integrable symbols.

In terms of applications, the Fredholm properties of Toeplitz operators play an important role. In particular, the fellow together with his PhD student A. Perala characterised the essential spectra of Toeplitz operators acting on Bergman spaces with matrix-valued symbols in two important classes: the Douglas algebra ("continuous plus analytic" functions) and the Zhu class (bounded functions of vanishing mean oscillation). We also obtained formulas for the Fredholm index in some cases, and conjectured what the index should be like in the most general case of these symbols classes. Our conjectures were recently verified by A. Perala and A. Bottcher. Our study has led to some new ideas that may allow one to study the Fredholm properties of Toeplitz operators with piecewise continuous symbols on Bergman spaces. We also wrote an article that reviewed the state of the art of Bergman space operators, provided a list of open problems together with some new results. This has generated new activity in the field and some progress related to boundedness, compactness and Fredholmness of these types of operators.

Project results

Hankel operators and matrices are closely related to orthogonal polynomials, Toeplitz matrices, Riemann-Hilbert problems and random matrix theory, which lead us to consider the theory of Hankel operators on weighted Fock spaces. We obtained characterisations for boundedness and compactness, but overall the results are not directly relevant to the project objectives.

The fellow also advised a PhD student at the State University of New York at Albany during the project. Their work is related to the use of Riemann-Hilbert problems in connection with spectral properties of Toeplitz operators. The results obtained include a description of the essential spectra of Toeplitz operators on Hardy spaces with certain continuous vanishing symbols (related to the oscillation of the symbols near their zeros). Regarding the theory of Riemann-Hilbert problems, we have considered situations where one is concerned with the asymptotic behaviour of RHPs with certain exponentially varying non-analytic data. Note that the Deift-Zhou nonlinear steepest descent method for such problems requires analyticity. We aim to extend the method to Riemann-Hilbert problems with non-analytic data (which is of interest in the theory of integrable systems), but no concrete results have yet been obtained.

Regarding the project's objectives, 1 and 2, the fellow has computed the formula for Toeplitz determinants with Fisher-Hartwig singularities using the Borodin-Okounkov formula. This is an interesting idea to deal with a problem that was previously solved via several different methods, such as the use of orthogonal polynomials and factorisations of Toeplitz matrices. Fisher-Hartwig symbols first appeared in connection with problems in statistical mechanics. More recently they have appeared in many diverse physical applications in random matrix theory, string theory and combinatorics. Our ongoing work, which started during the last phase of the project, is concerned with the asymptotics of Toeplitz determinants with transition-type singularities and their applications in random matrix theory using the powerful Riemann-Hilbert method and the Painleve transcendents. Part of the motivation comes from the connection to entanglement entropy and, in particular, we are dealing with questions related to entanglement in quantum spin chains and, as an application, the two-dimensional Ising model. Certain quantum phase transitions which occur in interacting lattice systems at zero temperature are of particular interest. Under these conditions the system is in the ground state (which is also a pure state), and any correlations must be a consequence of the fact that the ground state is entangled. Regarding expected concrete results, we deal with gliding singularities, and aim to obtain uniform asymptotics of Toeplitz determinants as singularities coalesce. In the future, we wish to study even more complicated (matrix-valued) singularities and consider their applications in random matrix theory and integrable systems; the ultimate goal is to obtain a complete classification of transition type singularities with uniform asymptotics of Toeplitz determinants with coalescing singularities.

It is worth noting that the asymptotics of orthogonal polynomials, random matrix theory and Riemann-Hilbert problems are very closely related to the properties of Toeplitz operators, and in particular the fellow has written a few articles in this direction, which somewhat deviates from Annex I but is quite closely connected to Objectives 1 and 2. A long standing problem is to find a complete characterisation of bounded (or compact) Toeplitz operators acting on certain function spaces, such as Bergman and Fock spaces. This difficult problem is present even in the Hilbert space setting. We have made good progress on these fundamental properties of Toeplitz operators generated by distributional symbols. In particular we showed that the membership of the symbols in a certain weighted Sobolev space is sufficient for the Toeplitz operator to be bounded, and showed the natural relation of the hyperbolic geometry of the disk and the order of the distribution. Similar results for compactness were also derived. It is worth noting that our results provided a large class of bounded and compact Bergman-space Toeplitz operators, whose symbols are far beyond bounded and even locally integrable symbols.

In terms of applications, the Fredholm properties of Toeplitz operators play an important role. In particular, the fellow together with his PhD student A. Perala characterised the essential spectra of Toeplitz operators acting on Bergman spaces with matrix-valued symbols in two important classes: the Douglas algebra ("continuous plus analytic" functions) and the Zhu class (bounded functions of vanishing mean oscillation). We also obtained formulas for the Fredholm index in some cases, and conjectured what the index should be like in the most general case of these symbols classes. Our conjectures were recently verified by A. Perala and A. Bottcher. Our study has led to some new ideas that may allow one to study the Fredholm properties of Toeplitz operators with piecewise continuous symbols on Bergman spaces. We also wrote an article that reviewed the state of the art of Bergman space operators, provided a list of open problems together with some new results. This has generated new activity in the field and some progress related to boundedness, compactness and Fredholmness of these types of operators.

Project results

Hankel operators and matrices are closely related to orthogonal polynomials, Toeplitz matrices, Riemann-Hilbert problems and random matrix theory, which lead us to consider the theory of Hankel operators on weighted Fock spaces. We obtained characterisations for boundedness and compactness, but overall the results are not directly relevant to the project objectives.

The fellow also advised a PhD student at the State University of New York at Albany during the project. Their work is related to the use of Riemann-Hilbert problems in connection with spectral properties of Toeplitz operators. The results obtained include a description of the essential spectra of Toeplitz operators on Hardy spaces with certain continuous vanishing symbols (related to the oscillation of the symbols near their zeros). Regarding the theory of Riemann-Hilbert problems, we have considered situations where one is concerned with the asymptotic behaviour of RHPs with certain exponentially varying non-analytic data. Note that the Deift-Zhou nonlinear steepest descent method for such problems requires analyticity. We aim to extend the method to Riemann-Hilbert problems with non-analytic data (which is of interest in the theory of integrable systems), but no concrete results have yet been obtained.