Random Matrices and Integrability in Complex and Quantum Systems

Research Workshop of The Israel Science Foundation

25 – 30 October 2023

[L-01] **Universality in the number variance and counting statistics of the real and symplectic Ginibre ensemble** / Gernot Akemann

We compute and compare the statistics of the number of eigenvalues in a centered disc of radius \( \, R \,\) in all three Ginibre ensembles.
We determine the mean and variance as functions of \(\, R \,\) in the vicinity of the origin, where the real and symplectic ensembles exhibit
respectively an additional attraction to or repulsion from the real axis, leading to different results. In the large radius limit,
all three ensembles coincide and display a universal bulk behaviour of \(\, {\mathcal O} (R^2)\,\) for the mean, and \(\, {\mathcal O} (R)\,\)
for the variance. We present detailed conjectures for the bulk and edge scaling behaviours of the real Ginibre ensemble, having real
and complex eigenvalues. For the symplectic ensemble we can go beyond the Gaussian case (corresponding to the Ginibre ensemble) and
prove the universality of the full counting statistics both in the bulk and at the edge of the spectrum for rotationally
invariant potentials, extending a recent work which considered the mean and the variance. This statistical behaviour
coincides with the universality class of the complex Ginibre ensemble, which has been shown to be associated with
the ground state of non-interacting fermions in a two-dimensional rotating harmonic trap. All our analytical results and
conjectures are corroborated by numerical simulations.

[L-02] **Constrained Mittag-Leffler ensemble in hard-edge scaling: Fluctuations of generic additive statistics** / Sergey Berezin

We consider the constrained Mittag-Leffler ensemble with particles confined to an origin-centered disk lying strictly
within the droplet. Applying the hard-edge scaling, we will focus on the region of size \( \, {\mathcal O}( 1/n) \,\) around the disk's boundary,
where \(\, n \,\) is the number of particles. We will discuss fluctuations of rotationally-invariant additive statistics arising from bounded
measurable functions and present the corresponding finite-dimensional CLT. This theorem extends an earlier result by
Y. Ameur, C. Charlier, J. Cronvall, and J. Lenells (2022) for the disk counting statistics. Further, we consider
statistics supported on the complement of a varying-radius origin-centered disk. These radius-dependent statistics
become continuous-time stochastic processes. We will present a functional limit theorem pertinent to this scenario and,
time permitting, discuss its consequences.

[L-03]
** The double scaled SYK model and non-commutative space-time** / Micha Berkooz

I will review the chord diagram technique for solving the double scaled Sachdev-Ye-Kitaev model (and many
other double scaled \( K\)-local radnom Hamiltonians), and show the emergence of a \( {\rm SL}_q (2) \) as a
"symmetry" of the model. One can then use this symmetry to construct a \( 2D \) non-commutative geometry
which is dual (in the sense of the AdS/CFT duality) to the ensemble average of the random Hamiltonians.

[L-04]
** Randomly pruning the Sachdev-Ye-Kitaev model** / Richard Berkovits

The Sachdev-Ye-Kitaev model (SYK) is renowned for its short-time chaotic behaviour, which plays a fundamental role
in its application to various fields such as quantum gravity and holography. The Thouless energy, representing the
energy scale at which the universal chaotic behaviour (RMT) in the energy spectrum ceases, can be determined
from the spectrum itself. When simulating the SYK model on classical or quantum computers, it is advantageous
to minimise the number of terms in the Hamiltonian by randomly pruning the couplings. In this talk, we
demonstrate that even with a significant pruning, eliminating a large number of couplings, the chaotic
behaviour persists up to short time scales. This is true even when only a fraction of the original
\(\, {\mathcal O}(L^4)\,\) couplings in the fully connected complex Fermion SYK model, specifically
\( \, {\mathcal O}(K\, L) \,\), is retained. Here, \( \, L \,\) represents the number of sites, and
\( \, K \sim 10 \,\). The properties of the long-range energy scales, corresponding to short time scales,
are verified through numerical singular value decomposition (SVD) and level number variance calculations
which mediate the effects of strong sample-to-sample fluctuations which make the unfolding of the spectrum
non-trivial.

[L-05] **Low complexity random matrices** / Eugene Bogomolny

A typical \( \, N \times N \,\) matrix requires about \( \, {\mathcal O} (N^3)\,\) operations to find, e.g., the inverse.
But there are classes of structured matrices that need at the most \( \, {\mathcal O} (N^2)\,\) operations. Such matrices,
called low complexity matrices, appear naturally in different domains. In particular, these are Toeplitz,
Hankel, and Toeplitz-plus-Hankel matrices as well as matrices corresponding to quantization of certain
pseudo-integrable billiards and quantum maps. When parameters of such matrices are chosen to be random,
one gets low-complexity random matrices. It appears that statistical properties of eigenvalues and
eigenfunctions of many such matrices differ considerably from the usual random matrix ensembles.
The talk is devoted to a discussion of these and related subjects.

[L-06] **Large deviations and fluctuations of real eigenvalues of elliptic random matrices** / Sung-Soo Byun

In this talk, I will discuss real eigenvalues of the elliptic Ginibre matrices, exploring both the strong
and weak non-Hermiticity regimes. In particular, I will present the central limit theorem and large deviation
probabilities for the number of real eigenvalues.

[L-07] **Many-body quantum chaos, spectral form factor, and Ginibre ensemble ** / Amos Chan

I will present three results on the universal aspects of many-body quantum chaotic systems that go
beyond the standard random matrix theory paradigm. Firstly, I will present an exact scaling form
of the spectral form factor (SFF) in a generic many-body quantum chaotic system, deriving the
so-called "bump-ramp-plateau" behaviour. Secondly, I will introduce and provide an analytical
solution of a generalization of SFF for non-Hermitian matrices, called Dissipative SFF, which
displays a "ramp-plateau" behaviour with a quadratic ramp. Thirdly, I will provide evidences
that non-Hermitian Ginibre ensemble behaviour surprisingly emerge in generic many-body quantum
chaotic systems, due to the presence of many-body interaction.

[L-08] ** The two-dimensional Coulomb gas: Fluctuations through a spectral gap** / Christophe Charlier

I will discuss various statistics for a class of radially symmetric Coulomb gas ensembles at inverse temperature
\( \,\beta=2\, \), for which the droplet consists of a number of concentric annuli. We typically find an
oscillatory behaviour in the distribution of particles which fall near the inner edges of the droplet.
These oscillations are given explicitly in terms of a discrete Gaussian distribution, weighted Szegö kernels,
and the Jacobi theta function.

[L-09] ** How do the eigenvalues of a large random matrix behave?** / Giorgio Cipolloni

We prove that the fluctuations of the eigenvalues converge to the Gaussian Free Field (GFF) on the unit disk.
These fluctuations appear on a non-natural scale, due to strong correlations between the eigenvalues.
Then, motivated by the long-time behaviour of the ODE \(\, \dot{u} = X\, u \,\), we give a precise estimate
on the eigenvalue with the largest real part and on the spectral radius of \(\, X \), and prove the
universality of their Gumbel fluctuations.

[L-10] ** The Rosenzweig-Porter model revisited for the three Wigner-Dyson symmetry classes ** / Barbara Dietz

Interest in the Rosenzweig-Porter model, a parameter-dependent random-matrix model that interpolates between
Poisson and Wigner-Dyson statistics, has come up again in recent years in the field of many-body quantum
chaos. The reason is that it exhibits parameter ranges in which the eigenvectors are Anderson-localized,
non-ergodic (fractal) and ergodic extended, respectively. We present numerical results for the fluctuation
properties in the eigenvalue spectra, comparison with existing analytical results and properties of the
eigenvectors in terms of the adiabatic gauge potential, which we recently applied to the mass-deformed S
achdev-Ye-Kitaev model, and Kullback-Leibler divergences. Here, we consider all symmetry classes of
Dyson's threefold way. Finally, I will present recent experimental results with a superconducting
microwave billiard undergoing a transition from a quantum system with integrable classical dynamics
to one with violated time-reversal invariance and chaotic classical counterpart.

[L-11] ** Supersymmetric quantum mechanics and a variational principle for orthonormal polynomials on the whole real line ** / Joshua Feinberg

A peculiar property of polynomials which are orthogonal with respect to some weight function along the
whole real line is that they do not obey a Sturm-Liouville type differential equation, with the
unique exception of Hermite polynomials. (This is in contrast with systems of polynomials orthogonal
on a compact or semi-compact segment.) By combining methods of supersymmetric quantum mechanics
and random matrix theory, we show that such polynomials obey a system of Hartree-Fock equations,
familiar from the theory of non-interacting Fermi gases. We demonstrate these equations for
the specific example of the continuous Hahn polynomials. These Hartree-Fock equations reduce to
the Schrödinger equation for the harmonic oscillator in the case of Hermite polynomials.

[L-12] ** Replica-symmetry breaking transitions in the large deviations of the ground-state of
a spherical spin-glass ** / Yan Fyodorov

The ground state (minimal energy) of the simplest \( p = 2 \) spherical spin glass coincides with the largest eigenvalue
of the GOE matrix, hence the associated Large Deviations (and, on a different scale, Tracy-Widom distribution) provide
a natural starting point for studying fluctuations of the minimal energy in a general spherical spin glass. We use
the replica method to develop a theory of Large Deviations of rate \( N \) for the general spherical spin glass
model with random magnetic field and reveal a surprisingly rich picture of Replica Symmetry Breaking (RSB) phenomena
required to describe such Large Deviations. Among other things, our results indicate that in spin glasses with
Full Replica symmetry breaking the typical fluctuations of the minimal energy should be described by new
distributions, different from Tracy-Widom. We then indentify the corresponding scaling exponents and tail behaviour.

[L-13] ** Universal dephasing mechanism of many-body quantum chaos ** / Victor Galitski

Both our everyday experience and laboratory experiments indicate that physical systems, initially
prepared in a non-equilibrium state, time-evolve into thermal equilibrium. Sets of axioms have been
formulated to justify this generic behavior and the emergence of statistical mechanics. Yet,
there is no clear understanding of the fundamental principles underlying thermalization and
ergodicity in many-body quantum systems. How do such irreversible thermal states emerge in
quantum systems, which are governed by the "reversable" unitary laws of quantum physics? This talk
will review recent work on this topic and the closely related field of many-body quantum chaos. The
concept of many-body quantum chaos will be defined and some its universal features will be derived
from first principles.

[L-14] ** Six-fold way of traversable wormholes in the Sachdev-Ye-Kitaev model ** / Antonio Garcia-Garcia

In the infrared limit, a nearly anti-de Sitter spacetime in two dimensions \( ({\rm AdS}_2) \) perturbed by
a weak double trace deformation and a two-site \( (q>2) \)-body Sachdev-Ye-Kitaev (SYK) model with
\( N \) Majoranas and a weak \( 2 r\)-body intersite coupling share the same near-conformal dynamics
described by a traversable wormhole. We exploit this relation to propose a symmetry classification
of traversable wormholes depending on \( N,\; q,\) and \(r,\) with \( q>2r,\) and confirm it
by a level statistics analysis using exact diagonalization techniques. Intriguingly, a time-reversed
state never results in a new state, so only six universality classes occur: A, AI, BDI, CI, C, and D.

[L-15] ** Symmetry breaking in random matrix approach to noncommutative geometries for quantum gravity ** / Sven Gnutzmann

Recently, John Barrett and coworkers introduced random noncommutative geometries in the context of finding
a mathematical approach to quantum gravity. The geometry is inscribed in a random Dirac operator and state
sum with a given free energy. The proposed form of a quartic free energy shows a phase transition at a
critical interaction. The model is equivalent to random matrix models that may contain a number of correlated
hermitian matrices. Even the simplest case where only one hermitian matrix is present shows interesting behaviour
such as symmetry breaking in the density of states. I believe that this approach may be of interest in the
random-matrix community. The aim of this talk is to review recent results with a focus on the single matrix
case and present recent analytical and numerical contributions obtained by Mauro D'Arcangelo
(PhD student of John Barrett) and myself. These results concern a Riemann-Hilbert approach to the density
of states in the matrix models where a phase transition and symmetry breaking can be observed.

[L-16] ** The Ising model coupled to 2D gravity ** / Nathan Hayford

One of the most celebrated exactly solvable models in statistical mechanics is the two-dimensional
Ising model. The original model, introduced in the 1920s, has a rich mathematical structure. It thus came
as a pleasant surprise when physicists studying matrix models of 2D gravity found that, coupled to
quantum gravity, the planar Ising model still had an elegant solution as a special instance
of the 2-matrix model. The methods used by V. Kazakov and his collaborators involved the method of
orthogonal polynomials. However, these methods were formal, and no direct analytic derivation of
the phase transition has been described in the literature since the original paper of V. Kazakov
in 1986. In this talk, we present a rigorous version of the Ising model coupled to 2D gravity,
using steepest descent analysis for biorthogonal polynomials. We find that the phase transition
is described by the string equation of the 3rd reduction to the KP hierarchy, in agreement with
the predictions of G. Moore, M. Douglas, and their collaborators.

[L-17] ** On the equipartition principle for Wigner matrices ** / Joscha Henheik

The total energy of an eigenstate in a composite quantum system tends to be distributed
equally among its constituents. In this talk, we explain how to identify the quantum
fluctuation around this Equipartition Principle in the simplest disordered quantum system
consisting of linear combinations of independent Wigner matrices. The proof is based on
a rigorous justification of the Eigenstate Thermalisation Hypothesis and Gaussian fluctuation
for general quadratic forms of the bulk eigenvectors of Wigner matrices with an arbitrary
deterministic deformation.

[L-18] ** Parisi's hypercube, SYK and black holes: Making a case for fluxed ensembles in Fock spaces ** / Yiyang Jia

We consider a model of Parisi where a single particle hops on an infinite-dimensional
hypercube, under the influence of a uniform but disordered magnetic flux. We reinterpret
the hypercube as the Fock-space graph of a many-body Hamiltonian, and the flux as a
frustration of the return amplitudes in Fock space. This model has the same Green's functions
as the Sachdev-Ye-Kitaev model and very similar random matrix level statistics. Hence it
is an equally good quantum model for a particular kind of black holes. We advocate for
the view that both models should be understood as examples of highly-fluxed ensembles in
Fock spaces, and discuss its possible connection to Berry phases.

[L-19] ** Random matrix theory and the information paradox ** / Mario Kieburg

The information paradox of black holes is the question about a thermal black hole radiation
while the evolution of a quantum system is unitary. Page has proposed a solution of this problem,
namely that the radiation only looks thermal because the quantum system of the particles that
are sent out are in general entangled with the quantum system describing the black hole.
Interestingly, Page's idea of a uniformly distributed pure state actually yields a random
matrix model for the reduced density matrix. In recent years, there has been some progress
on how typical this situation is, and extended Page's result to systems of Gaussian quantum
states with and without particle number conservation and for fermions as well as bosons. These
models are effectively embedded random matrix ensembles. Currently, Eric Aurell, Lucas Hackl,
Pawel Horodecki, Robert Jonsson and I study a random matrix model for Hawking's setting of
a bosonic scalar field. The model is novel in its nature as it is uniformly drawn from the
non-compact real symplectic group which is only constraint (and guaranteeing its normalisability)
by the symplectic eigenvalues of the individual modes emitted in the black hole radiation. The
question we address is the strength of correlation between the modes. In my talk, I will report
on this progress.

[L-20] ** Eikonal formulation of large dynamical random matrix models ** / Maciej Nowak

Standard approach to dynamical random matrix models relies on the description of trajectories
of eigenvalues. Using the analogy from optics, based on the duality between the Fermat principle
(trajectories) and the Huygens principle (wavefronts), we formulate the Hamilton-Jacobi
dynamics for large random matrix models. The resulting equations describe a broad class of random
matrix models in a unified way, including normal (Hermitian or unitary) as well as strictly non-normal
dynamics. HJ formalism applied to Brownian bridge dynamics allows one for calculations of the
asymptotics of the Harish-Chandra-Itzykson-Zuber integrals.

[L-21] ** Symmetry classification of Lindbladians ** / Tomaž Prosen

We discuss systematic symmetry classification of Lindblad superoperators describing general
(interacting) open quantum systems coupled to a Markovian environment. Our classification is based
on the behavior of the Lindbladian under antiunitary symmetries and unitary involutions. We find
that Hermiticity preservation reduces the number of symmetry classes, while trace preservation
and complete positivity do not, and that the set of admissible classes depends on the presence of
additional unitary symmetries: in their absence or in symmetry sectors containing steady states,
Lindbladians belong to one of ten non-Hermitian symmetry classes; if however, there are additional
symmetries and we consider non-steady-state sectors, they belong to a different set of 19 classes.
In both cases, we do not find classes with Kramer's degeneracy. While the abstract classification
is completely general, we then apply it to spin-1/2 chains. We explicitly build examples in all
ten classes of Lindbladians in steady-state sectors, describing standard physical processes such
as dephasing, spin injection and absorption, and incoherent hopping, thus illustrating the relevance
of our classification for practical physics applications. Finally, we show that the examples in
each class display unique random-matrix correlations. To fully resolve all symmetries, we
employ the combined analysis of bulk complex spacing ratios and the overlap of eigenvector pairs
related by symmetry operations.

[L-22] ** Edge and cusp mesoscopic eigenvalue statistics for Wigner-type matrices ** / Volodymyr Riabov

We establish universal Gaussian fluctuations for the mesoscopic linear eigenvalue statistics of
Wigner-type matrices at the edge and cusp singularities and the local minima of the limiting
spectral density. Subsequently, we identify a continuous one-parameter family of functionals
that govern the limiting bias and variance. We show that the variance is equivalent to a suitable
weighted \( {\dot{H}}^{1/2} \)-norm of the test function. Our analysis entirely covers both
transitionary regimes: between the square-root singularity at a regular edge and the
cubic-root singularity at a cusp, and between a sharp cusp and a non-zero local minimum
in the bulk of the limiting spectral density. Previously the mesoscopic eigenvalue statistics
were only studied in the bulk of the spectrum and near the regular square-root edges,
avoiding the cusp-like singularities.

[L-23] ** Many facets of the power spectrum ** / Roman Riser

The power spectrum has emerged as an effective tool for studying both system-specific and
universal properties of quantum systems. We will introduce the definition of the power-spectrum,
present its nonperturbative theory and confront our universal predictions with numerical
experiments involving the circular unitary ensemble and nontrivial zeros of the Riemann
zeta function. In the main part of the talk, we will explain how the power spectrum naturally
appears in several fundamental problems of the random matrix theory. These include a
nonperturbative and asymptotic analysis of (i) autocovariances of level spacings
in the \( {\rm Sine}_2 \) determinantal point process and (ii) statistics of local level spacings.

[L-24] ** High-dimensional random glassy landscapes: Two related random-matrix problems ** / Valentina Ros

High-dimensional random landscapes are typically very non-convex and glassy, with plenty of
stationary points (local minima, maxima or saddles). The local curvature of Gaussian
landscapes in the vicinity of their stationary points is described by matrices having GOE
statistics, deformed by both additive and multiplicative rank-1 perturbations. I will
discuss two properties of these perturbed GOE ensembles, namely (i) the joint large
deviation functions of their minimal eigenvalue and eigenvector, (ii) the typical
overlap between eigenvectors of pairs of such matrices, correlated with each others. I
will illustrate how these problem arise when addressing specific questions on the
random landscape, such as (i) what is the statistical distribution of saddle points
surrounding a given local minimum of the landscape, and (ii) what is the landscape
profile along paths interpolating between two of its local minima. The motivation for
these problems relies on the study of activated dynamics of glassy systems, as I
will briefly discuss.

[L-25] ** Symmetry classes of non-Hermitian quantum matter ** / Lucas Sá

We investigate the symmetries, correlations, and universality of non-Hermitian
many-body quantum systems (dissipative, Lindbladian, and PT-symmetric). We start by
considering a non-Hermitian \(q\)-body Sachdev-Ye-Kitev (nHSYK) model with \( N \) Majorana
fermions, relevant to the description of the short-time (or jump-free post-selected)
evolution of strongly-correlated quantum matter. Such systems can belong to 38 non-Hermitian
symmetry classes and we identify nine of them in the nHSYK model, depending on \( q \) and \( N \).
The level statistics on several timescales are well described by random matrix theory, however,
deviations at short times are identified and analytically characterized. Imposing the physical
constraints of Lindbladian or PT-symmetric evolution effectively doubles the degrees of
freedom but reduces the allowed set of symmetry classes, by preventing the existence of
classes with Kramers degeneracy. In particular, we show that simple (and experimentally
realizable) examples of Lindbladians in steady-state sectors belong to a tenfold set of
non-Ginibre classes and discuss the universality of their correlations.

[L-26] ** Dynamical manifestations of many-body quantum chaos within experimental reach ** / Lea Santos

The complex Fourier transform of the two-point correlator of the energy spectrum of a
quantum system is known as the spectral form factor. It is a useful diagnostic tool
of quantum chaos that functions even in the presence of symmetries. However, it offers
two drawbacks for experimental setups that focus on the quantum evolution of many-body
quantum systems, such as the experiments with cold atoms and with ion traps. The first
problem is that the properties of the spectral form factor are smeared out by large
temporal fluctuations, whose minimization requires disorder or time averages. This
requirement holds for any system size, because the spectral form factor is non-self-averaging.
We show that this issue can be solved by slightly opening the quantum system, which suppresses
quantum noise. The other problem is that, even though the spectral form factor provides the
means to study spectral correlations in the time domain, it is not a true dynamical quantity. We
discuss two genuine dynamical quantities that contain the spectral form factor in their long-time
dynamics. They are the survival probability (probability to find the evolved system in its initial
state) and the spin autocorrelation function, which can both be experimentally detected.

[L-27] ** Character expansion in non-Hermitian ensembles ** / Nicholas Simm

The archetypal model of a non-Hermitian random matrix is the Ginibre ensemble,
consisting of i.i.d. standard Gaussian entries with no symmetry constraints. Another
interesting non-Hermitian ensemble is obtained by truncating a Haar unitary matrix. I
will discuss the character expansion technique for evaluating correlations of
characteristic polynomials in such models. By employing the theory of symmetric
functions, particularly Schur and zonal polynomials, we give a unified treatment
in the real, complex and quaternion settings.

[L-28] ** How chaotic is a Hermitean matrix? ** / Uzy Smilansky

Given an \( N \times N\) Hermitian matrix \(H\), we derive an expression for the
largest Lyapunov exponent of the classical trajectories in the phase space
appropriate for the dynamics induced by \( H\). To this end, we associate to
\( H\) a simple graph with \( N \) vertices and derive a quantum map on
functions defined on the directed edges of the graph. Using the semi-classical
approach in the reverse direction, we obtain the corresponding classical
evolution (Liouvillian) operator. Using ergodic theory methods
(Sinai, Ruelle, Bowen, Pollicot.), we obtain close expressions for the
Lyapunov exponent as well as for its variance.

[L-29] ** Counting statistics for interacting fermions and random matrices ** / Naftali Smith

Over the past few decades, there have been spectacular experimental developments in manipulating cold
atoms (bosons or fermions), which allow one to probe quantum many-body physics, both for
interacting and noninteracting systems. I will begin by presenting results for the counting
statistics for a non interacting Fermi gas in \( d\) dimensions confined by a general trapping potential
(we assume a central potential for \( d>1\)), in the ground-state. In \( d=1\), for specific potentials,
this system is related to classical random matrix ensembles. I will then extend some of the results for
the \( d=1 \) case to fermions with interactions.

[L-30] ** Universal Chern number statistics in random matrix fields ** / Or Swartzberg

The discovery of the quantum Hall effect was followed by an observation that the Hall conductance
integer for systems with a band spectrum can be expressed as a Chern integer. In the lecture,
I will introduce the subject and show by example how Chern number enter into quantum mechanics.
I will present a work [O. Swartzberg, M. Wilkinson, O. Gat: SciPost Phys. **15,** 015 (2023)]
in which we investigate the probability distribution of Chern numbers for a parametric version of
the GUE random matrix ensemble, which is a model for a chaotic or disordered system. The
numerically-calculated single-band Chern number statistics agree well with predictions
based on the earlier study [O. Gat and M. Wilkinson: SciPost Phys. **10**, 149, (2021)] of
the statistics of the quantum adiabatic curvature, when the parametric correlation length
is small. However, contrary to an earlier conjecture, we find that the gap Chern numbers
are correlated, and that the correlation is weak but slowly-decaying. Also, the statistics of
weighted sums of Chern numbers differs markedly from predictions based upon the hypothesis that
gap Chern numbers are uncorrelated. All our results are consistent with the universality hypothesis
described in the earlier paper, including in the previously unstudied regime of large correlation
length, where the Chern statistics is highly non-Gaussian.

[L-31] ** Topological constraints on optimal uniform approximations of random normal matrices ** / Razvan Teodorescu

In the random matrix theory models for interacting spin lattices, an important quantity describing
fluctuations around equilibrium is the emptiness formation probability. We generalize this problem to
include operators with complex spectra, by considering the formation of a planar fluctuation \( {\mathcal D}\)
in the support of the spectrum of a random normal operator. The partition of the plane into the domain \( {\mathcal D}\)
and its complement is then naturally characterized by the uniform distance between the antiholomorphic coordinate
supported on \( {\mathcal D}\) and the set of rational holomorphic functions supported on the complement,
leading to the question of optimal fluctuations, which minimize this distance: what are the connectivity
and the conformal class of the most likely (i.e., optimal) fluctuation away from equilibrium? This
talk answers both questions in full.

[L-32] ** Introduction to the Sachdev-Ye-Kitaev model ** / Jac Verbaarschot

The Sachdev-Ye-Kitaev (SYK) model has become the standard model of many-body quantum chaos
exhibiting universal level correlations, eigenstate thermalization as well as maximal chaos.
In the past couple of years, it has received a great deal of attention because of its connection
with two-dimensional quantum gravity. In this lecture, we will give an overview of the main
properties of the SYK model and its close relationship with Random Matrix Theory.
We emphasize the distinction between single-particle Hamiltonians and many-body systems
and discuss the mathematical structure underlying the solvability of this model. We close
with some recent developments such as the application of the SYK model to open quantum
systems and constraints on non-Hermitian many-body systems.

[L-33] ** Biorthogonal ensemble related to disordered wires and the universal conductance fluctuations ** / Dong Wang

The random matrix theory of quantum transport was set up thirty years ago, especially in dimensions \( 0 \) and \(1 \). The
transport problem for one-dimensional disordered wires can be modelled by the Dorokhov-Mello-Pereyra-Kumar (DMPK) equation
that is similar to the Dyson Brownian motion, and if the time-reversal symmetry is broken, the DMPK equation has a free
fermion solution. After taking the metallic limit, the free fermion solution becomes a biorthogonal ensemble.
The biorthogonal ensemble has the form
$$
\frac{1}{Z_n} \prod_{1\le i < j \le n} (x_i -x_j) \left(
f(x_i) - f(x_j)
\right) \prod_{i=1}^n x_i^\alpha h(x_i) {\rm exp} \left(
- n V(x_i)
\right),
$$
where \( f(x) = \sinh^2 (\sqrt{x}) \) and \( V(x) \) is a linear function. Based on this biorthogonal ensemble, the
"universal conductance fluctuation" property of disordered wires can be established, although a rigorous proof is still
in want of. The biorthogonal ensemble is a determinantal point process, and the correlation kernel is expressed
by biorthogonal polynomials. In this talk, we give the Plancherel-Rotach type asymptotics of the
biorthogonal polynomials, which lead to a rigorous proof of the universal conductance fluctuation result from
the biorthogonal ensemble. Our approach is via the vector Riemann-Hilbert problem.

[L-34] ** Averages of products of characteristic polynomials and the law of real eigenvalues
for the real Ginibre ensemble ** / Oleg Zaboronski

An elementary derivation of the Borodin-Sinclair-Forrester-Nagao Pfaffian point process, which
characterises the law of real eigenvalues for the real Ginibre ensemble in the large matrix size
limit, uses the averages of products of characteristic polynomials. This derivation reveals a
number of interesting structures associated with the real Ginibre ensemble such as the hidden
symplectic symmetry of the statistics of real eigenvalues and an integral representation for the
\(K\)-point correlation function for any \( K \in {\mathbb N} \) in terms of an asymptotically exact
integral over the symmetric space \( {\rm U}(2K)/{\rm USp}(2K)\).

[P-01] **Wegner model on a tree graph: \( {\rm U}(1)\) symmetry breaking and a non-standard phase of disordered electronic matter** / Julian Arenz

Assuming the self-consistent theory of localization due to About-Chacra et al., we solve the \( N=1 \) Wegner
model in the regime of strong disorder and high dimension. In the process, we uncover a non-standard electronic
phase with spontaneously broken \( {\rm U} (1) \) symmetry − the missing field-theory basis underlying phenomena
associated with fractal eigenstates and singular continuous spectra.

[P-02] **Norm convergence rate for polynomials of random matrices ** / Jacob Fronk

We study a general class of Hermitian non-commutative quadratic polynomials of multiple independent
Wigner matrices. We establish that, as the dimension \( N \) of the matrices grows to infinity, the
operator norm of such polynomials \( q \) converges to a deterministic limit and we prove an optimal
rate of convergence of \( N^{-2/3+o(1)}\). To obtain the result, we study the limit of the eigenvalue
density of \( q\). We prove that this density always has a square root growth at its edges as well
as an optimal local law around the edges. Combining the two results leads to the desired rate of
convergence.

[P-03] **Statistics of local level spacings in quantum chaology** / Peng Tian

We introduce a notion of local level spacings and study their statistics within a
random-matrix-theory approach. In the limit of infinite-dimensional random matrices, we
identify the two universal classes of local spacings distributions which describe unfolded
spectra of quantum systems with fully chaotic and completely integrable classical dynamics,
respectively. We further argue, and explicitly demonstrate by exact diagonalisation of the
Sachdev-Ye-Kitaev (SYK) Hamitonians, that the ratios of averaged local spacings computed
for raw spectra maintain their universality thus offering a framework to monitor spectral
statistics in quantum many-body systems.

[P-04] **Large deviations of spectral linear statistics of Gaussian random matrices** / Aleksandr Valov

We evaluated, in the large-N limit, the complete probability distribution \( {\mathcal P}(A,m)\) of the values
\( A\) of the sum \(\sum_{j=1}^N | \lambda_j |^m\), where \( \lambda_j \) \( (j=1,..,N)\) are the eigenvalues of
a Gaussian random matrix, and \( m \) is an arbitrary positive number. We employed the Coulomb gas method
and also performed numerical simulations by using a variant of the Wang-Landau algorithm. We found that
the rate function of \( {\mathcal P}(A,m)\) exhibits phase transitions of different characters on
the \( (A,m)\)-plane. The phase diagram of the system on this plane is surprisingly rich. It
consists of three regions: (i) a region with single-interval support of the eigenvalues spectrum,
(ii) a region emerging for \( m<2 \) where the spectrum splits into two separate intervals,
and (iii) a region emerging for \( m>2\), where the maximal or minimal eigenvalue "evaporates" from the
rest of eigenvalues. The phase transition between regions (i) and (iii) is of the second order.
To clarify the transition character between regions (i) and (ii), we focused on \( m=1\). Here, the transition
appears to be of infinite order. Remarkably, this transition, as well as the transition between
the regions (i) and (iii), occurs exactly at the ground state of the Coulomb gas. Our analytical
results are fully supported by the numerical simulations.

Last updated: October 06, 2023

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