Abstracts and Presentations

You may download the abstracts as a PDF file .

Topical session "Random matrices and ideas of integrability"

The function theory of matrix integrals, with applications / Mark Adler   →   presentation file 

We will discuss the technique of deforming random matrix integrals by deforming the measures so as to produce functions of integrable systems and then using Virasoro identities so as to produce partial differential equations for various probabilities. We will apply this to some older examples.

Integrability and random matrices / Eugene Bogomolny   →   presentation file 

A new class of random matrix ensembles is proposed. Random matrices of these ensembles are Lax matrices of classically integrable systems with a certain distribution of momenta and coordinates. The existence of integrable structure permits to calculate the joint distribution of eigenvalues of these matrices analytically. Spectral statistics of these ensembles are quite unusual in many cases. In particular, it appears that random matrices considered in PRL 93, 254102 (2004) are related with the Ruijsenaars model and this identification allows to obtain rigorously new examples of intermediate statistics.

Painlevé transcendents and their appearance in physics and random matrices / Alexander Its   →   presentation file 

Painlevé functions have been playing increasingly important role in random matrix theory since the early nineties works on the 2D quantum gravity of Brézin and Kazakov, Douglas and Shenker, and Gross and Migdal, and the works (also early nineties) of Mehta and Mahoux, and Tracy and Widom on the level spacing distributions. Indeed, the appearance of Painlevé functions in the field can be traced to 1970s−1980s works of Barouch, McCoy, Tracy, and Wu, and of Jimbo, Miwa, Môri and Sato on the quantum correlation functions. In this talk we will try to review these, and also some of the more recent results concerning Painlevé transcendents and random matrices. We will present a unified point view on the subject based on the Riemann-Hilbert method. The emphasis will be made on the various double scaling limits related to the universal properties of random matrices for which Painlevé functions provide an adequate "special function environment".

Painlevé transcendents and quantum transport / Eugene Kanzieper   →   presentation file 

In this talk, I will show that the paradigmatic problem of conductance fluctuations in chaotic cavities with broken time-reversal symmetry is completely integrable in the universal transport regime. This observation will be utilised to prove that the cumulant generating function of the Landauer conductance in the cavities probed via ballistic point contacts is given by the fifth Painlevé transcendent. If time permits, a closely related integrable theory of the noise power fluctuations in the crossover regime between thermal and shot noise will also be outlined.

Dyson's non-intersecting Brownian motions and multi-component KP-hierarchy / Pierre van Moerbeke   →   presentation file 

In the early 60's, Dyson introduced dynamics in random matrices and was led to non-intersecting Brownian motions on the real line. Scaling limits of these non-intersecting Brownian motions, when the number of particles tends to infinity, has led to a number of critical infinite-dimensional diffusions, much inspired by random matrix theory. Their transition probabilities are described by the Fredholm determinant of kernels (and extended kernels in the multi-time case), but also satisfy non-linear partial differential equations, obtained by scaling limits of the multi-component KP-hierarchy, combined with Virasoro constraints. I will give a descriptive overview of the different critical diffusions arising in this context.

The Sakai scheme−Askey table correspondence and determinantal point processes / Nicholas Witte   →   presentation file 

Key averages arising in random matrix theory have an evaluation as a determinant or multi-dimensional integral with a weight that is a natural deformation of the classical weights − the Hermite, Laguerre and Jacobi weights. These are known, in their most general settings, to be functions of the continuous Painlevé equations PVI , PV and PIV for the finite rank JUE/CyUE, LUE and GUE, and PV, PIII, PII for the bulk, hard and soft edge regimes. We report on a program to extend this understanding beyond the classical weights to the full Askey table, which encompasses all the hypergeometric and basic hypergeometric orthogonal polynomial systems. This involves the calculation of analogs to the traditional averages, which are usually defined on discrete and possibly non-uniform lattices. The integrable systems that are relevant here are those in Sakai Scheme − a hierarchical scheme containing a master elliptic Painlevé equation, a category of − Painlevé equations and another of discrete Painlevé equations which have the continuous Painlevé equations as limiting cases. The probabilistic models that these analog averages arise from are determinantal point processes which generalise random matrix ensembles and include exclusion processes, growth models, queuing theory and tiling problems. We will survey the currently available evidence for this correspondence.

Topical session "Replicas and integrability"

Intersection theory from duality and replica / Shinobu Hikami   →   presentation file 

Kontsevich’s work on Airy matrix integrals has led to explicit results for the intersection numbers of the moduli space of curves. In this article we show that a duality between − point functions on matrices and −point functions of    matrices, plus the replica method, familiar in the theory of disordered systems, allows one to recover Kontsevich’s results on the intersection numbers, and to generalize them to other models. This provides an alternative and simple way to compute intersection numbers with one marked point, and leads also to some new results. This is a joint work with E. Brézin.

Integrable theory of characteristic polynomials, with applications / Vladimir Al. Osipov   →   presentation file 

Ideas of integrability are used to build a non-perturbative theory of correlation functions of characteristic polynomials arising in the context of Hermitian unitary invariant random matrix ensembles. In particular, a technology to generate nonlinear differential equations satisfied by these correlation functions is presented. This approach is further used to demonstrate exact integrability of replica field theories in zero dimensions.

Replica quarks / Kim Splittorff   →   presentation file 

This talk discusses applications of the replica trick in QCD. Here the replica index counts the number of replicated quark flavors. In the integrable microscopic domain of QCD such replica quarks have uncovered the relationship between the spectral density of the Dirac operator and the spontaneous breaking of chiral symmetry.

Replicas, Toda lattice equations, and QCD / Jac Verbaarschot   →   presentation file 

In the case of the absence of anti-unitary symmetries, random matrix partition functions for different number of replicas are related by the Toda lattice equation. This relation makes it possible to express quenched averages in terms of the product of a bosonic and a fermionic partition function. In addition to explaining factorisation properties of known results, this method has enabled us to derive the quenched microscopic spectral density for QCD at nonzero chemical potential which was not known previously. In this lecture we explain the origin of the integrable structure and its extension to nonhermitean chiral random matrix theories for QCD at nonzero chemical potential. Applications to lattice QCD will be discussed as well.

Topical session "Growth processes, integrable systems, and non-Hermitean random matrices"

From random matrices to evaporating droplets / Oded Agam   →   presentation file 

The eigenvalues of normal random matrices, with the appropriate statistical weight, occupy a domain in the complex plain, which may be associated with the boundary of a Saffman-Taylor (ST) bubble in Hele-Shaw cell. Such a bubble is formed when a fluid of low viscosity (say air) is rapidly injected into the center of a cell built from two parallel plates where the narrow gap between them is filled with a highly viscous fluid (say oil). The area of the bubble is proportional to the number of eigenvalues, thus by taking the appropriate limit of the random matrix model, one can construct solutions describing the evolution of ST bubbles as their area increases. However, unlike the traditional ST problem, some of these solutions represent situations where new bubbles may appear, merge, or disappear. Nevertheless, this kind of behavior has been observed in evaporating droplets made of thin water films on a mica substrate. Due to competition between van der Waals and polar surface forces, such a film undergoes a first order phase transition between two values of its thickness, and by inducing a finite evaporation rate of the water, the interface between the two phases develops a fingering instability similar to that observed in the ST problem. Yet, evaporating droplets also experience a Rayliegh instability along their perimeters, and this instability selects only a special set of solutions.

Gap probabilities in non-Hermitean random matrix theory / Gernot Akemann   →   presentation file 

We compute the gap probabilities and individual eigenvalue distributions with respect to radial ordering. Four different symmetry classes of Ginibre and chiral type are compared, and the corresponding quantities are given is terms of a Fredholm determinant or Pfaffian for and , respectively. For rotational invariant weights the Fredholm eigenvalues for and are shown to be related, and we give most explicit expressions for the Gaussian chiral ensembles. Then the asymptotic expansion and level repulsion is discussed in all four ensembles. Finally exact and approximate results are compared to individual complex Dirac operator eigenvalues from Lattice solutions of QCD, the theory of strong interactions with finite quark density.

Multifractal spectra of critical curves / Eldad Bettelheim   →   presentation file 

The multifractal spectra of critical curves in 2D (such as cluster boundaries in critical stat-mech systems) is computed using methods of conformal field theory.

The real Ginibre ensemble: An integrable structure / Hans-Jürgen Sommers   →   presentation file 

For the real Ginibre ensemble of Gaussian real asymmetric matrices with orthogonal symmetry we present the joint probability density of real and pairwise complex conjugate eigenvalues. From this all −point densities in the complex plane are calculated as Pfaffians. The eigenvalues are concentrated in a disk of radius and with a finite fraction exactly on the real axis. They show cubic repulsion in the complex plane and linear repulsion on and from the real axis. It is shown that a very simple skew symmetric analytic kernel in general dimension or equivalently a simple tridiagonal skew symmetric matrix , which both have in all even dimensions the same form, determines all correlation functions and moments. The kernel can be obtained by Edelman's expression for the density of complex eigenvalues, but independently from a supersymmetric calculation of an average of a product of two characteristic polynomials. We will give an explicit expression for averages of all Schur functions of the random matrix considered. This allows for calculating averages of any analytic symmetric function of eigenvalues by Schur function expansions.

Stochastic growth models and matrix-valued diffusions / Herbert Spohn   →   presentation file 

On large scales one-dimensional ballistic growth with curved substrate has the same statistics as the edge line of GUE matrix-valued diffusions. While intriguing, there are other geometries, respectively initial conditions for growths models for which no random matrix analogue seems to be available. In my talk I will cover both aspects.

The asymmetric simple exclusion process: Integrable structure and limit theorems / Craig A. Tracy   →   presentation file 

Since its introduction nearly forty years ago, the asymmetric simple exclusion process (ASEP), has become the "default stochastic model for transport phenomena". Some have called the ASEP the "Ising model for nonequilibrium physics". In ASEP on the integer lattice particles move according to two rules: (1) A particle at waits an exponential time with parameter one (independently of all the other particles), and then it chooses with probability ; (2) If is vacant at that time it moves to , while if is occupied it remains at and restarts the clock. The adjective "simple" refers to the fact that allowed jumps are one step to the right, , or one step to the left, . The asymmetric condition means so that there is a net drift to either the right or the left. In this lecture we consider ASEP on the integer lattice with step initial condition: At time zero the particles are located at and there is a drift to the left (). If denotes the position of the th particle from the left at time (so that ), a basic quantity is the distribution function which describes the "current fluctuations". Physicists have conjectured that the limiting distribution of as with fixed is in the Kardar-Parisi-Zhang Universality Class. We show that this is indeed the case and describe the limiting distribution function. (This limiting distribution function first appeared in the random matrix theory literature.) This result extends an earlier theorem of Kurt Johansson on the T(totally)ASEP where and . This work is joint work with Harold Widom.

2D Dyson diffusion and growing patterns / Paul Wiegmann

At equilibrium particles followed Dyson diffusion in 2D in a harmonic confining potential uniformly fill a disk. A number of important problems in hydrodynamics, non-equilibrium driven processes, theory of orthogonal polynomials, soliton theory, etc, lead to a 2D Dyson diffusion in a potential perturbed by higher harmonics. These potentials are not confining: there are direction where particles can escape to infinity. In this case, however, there are steady distributions with a flux when particles remain confined. Their distribution is no longer two-dimensional. It consists of a 2D domain and a branching graph of curved lines. Topology of the graph is changing with an increasing number of diffusing particles. Singular 1D distribution represents shock fronts in certain hydrodynamic problems, lines of growth in solidification processes and also distribution of zeros of orthogonal polynomials and also distribution of eigenvalues in random matrices.

Dyson gas simulation of growing patterns: Geometry and integrability / Anton Zabrodin   →   presentation file 

We show how growth processes of Laplacian type can be simulated by statistical mechanics of 2D Coulomb charges in an external field (the Dyson gas), which may be thought of as eigenvalues of normal or complex random matrices. The growing cluster is represented by a domain where the mean density of the charges does not vanish in the large limit, and the physical growth time is identified with a coupling constant of the external field. The Dyson gas picture applies both to Laplacian growth of smooth domains in the plane and to the growth of slit domains described by the Loewner equation. It also provides a key to integrable structure of the models and gives a unique way of their discretisation or "quantization" preserving integrability. From this point of view, we discuss growth models associated to the Toda lattice and KP integrable hierarchies.

Topical session "Heuristic replica applications in random matrix theory and beyond"

Extreme value statistics in models with logarithmic correlations / Yan Fyodorov   →   presentation file 

The covariance of the Gaussian two-dimensional free field depends logarithmically on the distance. Using the methods of statistical mechanics of disordered systems we study the distribution of the global minimum of such a field sampled along some one-dimensional curves. The procedure relies, in particular, upon a replica-like continuation of Selberg integrals to negative dimensions. The presentation will be based on the joint works with Jean-Philippe Bouchaud and Pierre Le Doussal.

Universality class of replica symmetry breaking and its relation to growth phenomena / Reinhold Oppermann   →   presentation file 

Replica symmetry breaking (RSB) of the Sherrington–Kirkpatrick (SK) spin glass is elaborated as a pseudo−1D theory with two critical points at zero temperature and zero field. A "finite-size" scaling theory on the pseudo-lattice of RSB-orders is constructed and controlled by high order solutions (up to 200 RSB-steps) of a low- regularised self-consistent scheme. The scaling regime of small , small magnetic fields , small temperatures is analysed which leads to an understanding of the non-commuting limits and . The order function (here, replaces Parisi's ) is obtained as a fixed-point function under RSB-flow, being interpreted as a pseudo-time, and the critical points identified at and at . The critical points and separate (i) nonanalytic −  from −dependence and (ii) continuous from discontinuous parameter spectra. The so-called continuum limit is not fully continuous. The break point criterion is obtained as a forbidden (order parameter) level crossing. Crossover lines separating 1D-critical RSB−  from mean-field SK-behavior obey the power laws and at the two critical points, respectively. Further power laws are reported, distinguishing the universality class of RSB in the SK-model from similar cases, like KPZ interface growth, directed polymers etc. An analogy with linear and nonlinear fixed points of the KPZ-equation is explained. A partial differential equation, derived for the (continuous section of the) limit, is transformed into a Burgers-type equation with coefficients depending on pseudo-time , which prevents a linearisation by means of the Cole-Hopf transformation. Upgraded by means of a noise-term a new KPZ-type equation is defined and its renormalisation is considered. Two independent goals of the present theory have been to render the Parisi-phase comparable with Fisher-Huse droplet theory at and to provide simple analytical forms for the regularised Parisi order function. The relation to fluid mechanics and to the Le Doussal-Wiese functional RG of RSB have been a by-product.

Topical session "Novel field-theoretic approaches for random matrices and disordered systems"

Bosonisation for random matrices and electron systems in arbitrary dimensions / Konstantin Efetov   →   presentation file 

We suggest a general scheme based on the supersymmetry technique that allows one to consider low lying excitations in electron systems. The same approach allows one to consider correlations with close eigenvalues in the theory of random matrices. The effective field theory obtained in this scheme is a generalisation of the supermatrix sigma-model and can be derived exactly. It is demonstrated that a supersymmetric field theory can also be derived for systems of interacting electrons.

Propagation of a wave packet in the presence of non-linearity and random scatterers / Alexander Finkelstein

Recently, there appeared experiments and also numerics in which the non-linear Schrödinger equation and/or Gross-Pitaevskii equation have been studied in the presence of disorder. The relevant experiments are on photonic crystals and also on cold atoms. Therefore there is a need for non-linear equations describing propagation of the injected pulse (in the case of photonics) or the released atomic cloud (in the case of cold atoms). We derived a system of non-linear equations describing the energy distribution and the profile of the laser beam in the photonic crystal (or the density distribution in the case of the atomic cloud). The derivation has been performed either by finding a saddle point equation for the corresponding nonlinear sigma-model or, equivalently, by using quasi-classical Green's functions. A physical interpretation of rather non-trivial phenomena in the case of repulsive or attractive signs of the non-linearity is suggested.

Horizons in random matrix theory, Hawking radiation, and flow of cold atoms / Fabio Franchini   →   presentation file 

Standard invariant random matrix ensembles (RME) are characterised by extended eigenstates and Wigner semicircle law for the distribution of eigenvalues. In this work, we consider invariant RME with weak confinement (confinements weaker than any polynomial): as the confinement coupling increases, the level statistic changes from Wigner-like to progressively more Poisson-like, which normally would indicate a chaotic, localised system. We propose a low-energy field theory description for such a weakly confined ensemble, in terms of a free bosonic field propagating in a curved space-time characterized by the presence of an Event Horison. The construction of this effective theory is based on the identification of the two-point correlation function for the theory. In the limit of no confinement, the space-time becomes flat as one expects. The existence of an Event Horison in the curved case leads directly to the appearance of the phenomenon known as Hawking radiation: we discuss its physical interpretation in the original RME. This approach is similar in spirit to one recently developed to describe a trans-sonic Bose-Einstein condensate.

Almost diagonal random matrices: Application to multifractal metal and insulator / Vladimir Kravtsov   →   presentation file 

We review the recently developed formalism of "virial expansion" in a number of resonant energy levels which gives expansion of various correlations functions for Gaussian, non-invariant ensembles with parametrically small off-diagonal matrix elements in powers of this small parameter. The formalism is applied to a special type of banded matrix ensemble which gives a qualitative description of critical level and eigenfunction statistics at the Anderson transition point in 3D space and also allows an extension to Anderson insulator which contains both the critical and Mott's variable range hopping physics.

Superbosonisation meets free probability / Martin Zirnbauer   →   presentation file 

We sketch a proof of universality of local level correlation functions for non-Gaussian invariant random matrix ensembles, by using a method based on the so-called superbosonisation formula in combination with elements of free probability theory. Superbosonisation, a variant of the method of commuting and anticommuting variables, eclipses the traditional Hubbard-Stratonovich transformation in that it is not restricted to Gaussian probability distributions. In this talk we consider random matrices distributed according to a probability measure of the form with being a polynomial. To apply the superbosonisation formula, one needs to have control of the Fourier transform of the measure in the limit of infinite matrix size . We show this Fourier transform to be determined by a key notion in free probability theory: the Voiculescu −transform of the asymptotic level density.

Topical session "Towards nonperturbative theories of many-body systems"

Dynamic correlation functions of 1D quantum liquids / Alex Kamenev   →   presentation file 

I shall describe a recent progress in understanding of energy and momentum resolved dynamic correlation functions of 1D quantum liquids. In particularly, it was realised that there are specific lines (modes) on the energy-momentum plane, where the correlation functions exhibit power-law singularities. The exponents at these singularities depend on momentum and may be computed exactly for simple models. I shall discuss the physical origin of the singularities and the ways they may be observed.

Supersymmetry for disordered systems with interaction / Georg Schwiete   →   presentation file 

A scheme that allows to include electron-electron interaction into a supermatrix sigma-model for disordered electron systems is discussed. The method is based on replacing the initial model of interacting electrons by a fully supersymmetric model. Although this replacement is not exact, it is a good approximation for a weak short range interaction and arbitrary disorder. The replacement makes the averaging over disorder and further manipulations straightforward and leads to a supermatrix sigma-model containing an interaction term. The model has been studied in perturbation theory and renormalisation group calculations. The renormalisability of the model in the first loop approximation and in the first order in the interaction can be checked. In this limit it reproduces the renormalization group equations known from earlier works.

Poster session A

Decomposition formulae in matrix models / Alexander Alexandrov   →   presentation file 

Various branches of matrix model partition function can be represented as intertwined products of universal elementary constituents: Gaussian partition functions and Kontsevich functions . Technically, decomposition formulas are related to representation theory of algebras of Krichever-Novikov type on families of spectral curves with additional Seiberg-Witten structure. Representations of these algebras are encoded in terms of "the global partition functions". They interpolate between and associated with different singularities on spectral Riemann surfaces. This construction is nothing but M-theory-like unification of various matrix models with explicit and representative realisation of dualities.

A Lagrangean formalism for Hermitean matrix models / Alexander Klitz   →   presentation file 

Eynard's formulation of Hermitean one-matrix models in terms of intrinsic quantities of an associated hyperelliptic Riemann surface is rephrased as a Lagrangean field theory of a scalar particle propagating on the hyperelliptic surface with muliple self-interactions and particle-source interactions. Both types of interaction take place at the branch points of the hyperelliptic surface.

Construction of quantum integrable anyonic lattice and field models / Anjan Kundu   →   presentation file 

Quantum integrability of 1D anyonic models, which remained as a challenging problem due to its non-ultralocal nature (non-commutativity at space-separated points), is solved completely through the braided extension of the Yang-Baxter relation. Using two differnt realisations of the underlying generalised anyonic algebra, we construct two exactly solvable lattice models: (i) a hard-core anyon model with nearest-neighbor interaction and (ii) a lattice version of the anyonic nonlinear Schrödinger equation (NLS). Both these models allow exact algebraic Bethe ansatz solution. The later model at the continuum limit goes to a novel quantum integrable anyonic NLS field model, which in the −particle sector yields the exactly solvable −function interacting anyon gas, proposed earlier by the author. Quantum deformation of the above scheme is hoped to lead to a promising anyonic representation of the quantum group and a quantum deformation of the anyonic models. This should yield in particular an integrable anyonic derivative NLS model, producing exactly solvable derivative −function interacting anyonic gas.

The super-integrable Toda models / David Schmidtt   →   presentation file 

We derive the off-shell super-integrable extension of the Leznov-Saveliev (LS) equations. The on-shell derivation is obtained by solving an extended Riemann-Hilbert factorisation problem while the off-shell one is obtained by gauging a 2-Loop WZNW model. The two approaches provide in a natural way a new set of algebraic constraints for the existence of local supersymmetric flow in the supersymmetric affine Toda field theories (s-ATFT) associated to the LS equations. The definition of involves fermion bilinears coupled to Lie-algebra bosonic generators. We propose a solution for these equations encoding an interesting algebraic structure which is also explored. Some known supersymmetric Toda models are recovered, a new bosonic model which truly interpolates between the sinh-Gordon and sine-Gordon models is introduced. In the case when we introduce a new family of fermionic integrable models. In the simplest case, the construction reduces to a generalized Thirring-type integrable model.

Poster session B

Breaking of time-reversal symmetry of elastic waves in chaotic cavity by amplified feedback loop / Oleg Antoniuk   →   presentation file 

Elastic wave propagation in many materials obeys time reversal (TR) symmetry and reciprocity. These fundamental symmetries influence the statistics of the resonance frequencies in chaotic systems. The absence of TR symmetry should influence nearest neighbor resonance spacing distribution according to random matrix theory, e.g. by reducing amount of small (relative to the average) frequency spacings between the resonances. We perform ultrasonic experiments in aluminum chaotic cavities with feedback. It is achieved by connecting an amplified loop between two sensors on the surface of the cavity. This loop destroys reciprocity and TR symmetry and therefore influences statistical properties in a controlled way. We present intensity distributions, nearest neighbor resonance spacing distributions and spectral rigidity for different feedback levels. We also present results of time reversal experiments. These experiments show explicitly how efficient the excitation pulse can be reconstructed by replaying back in time a track of cavity oscillations recorded at certain delay after exciting pulse (the oscillations that arrive first are released last). Efficiency of refocusing of original excitation pulse is also shown to be dependent on amplification coefficient in the feedback loop breaking the reciprocity and TR symmetry.

The partition function of a multi-component Coulomb gas on a circle / Niko Jokela and Matti Järvinen   →   presentation file 

We study a two-dimensional Coulomb gas consisting of a mixture of particles carrying various positive multiple integer charges, confined on a unit circle. We consider the system in the canonical and grand canonical ensembles, and attempt to calculate the partition functions analytically, using Toeplitz and confluent Vandermonde determinants. Just like in the simple one-component system (Dyson gas), the partition functions simplify at special temperature , allowing us to find compact expressions for them.

The correspondence between Tracy-Widom and Adler-Shiota-van Moerbeke approaches in random matrix theory / Igor Rumanov
→   presentation file 

Two approaches (TW and ASvM) to derivation of integrable differential equations for random matrix probabilities are compared. Both methods are rewritten in such a form that simple and explicit relations between all TW dependent variables and functions of ASvM are found, for all unitary ensembles of random matrices. Orthogonal function systems and Toda lattice are seen at the core of both approaches and their relationship. Moreover, Toda-AKNS system provides a universal structure of partial differential equations for all unitary ensembles. Interestingly, this structure can be seen in two very different forms: one arises from orthogonal functions−Toda lattice considerations, while the other comes from Schlesinger equations for isomonodromic deformations and their relation with TW equations. The originally considered example of finite size Gaussian matrices most neatly exposes this structure.

   Please contact the workshop organisers at Send email if you have any further questions.