Matrix Addition and the Dunkl Transform at High Temperature

In recent decades, determinants and Pfaffians were found for eigenvalue correlations of various random matrix ensembles. These structures simplify the average over a large number of ratios of characteristic polynomials to integrations over one and two characteristic polynomials only. Up to now it was thought that determinants occur for ensembles with Dyson index β = 2, whereas Pfaffians only for ensembles with β = 1, 4. We derive a non-trivial Pfaffian determinant for β = 2 random matrix ensembles which is similar to the one for β = 1, 4. Thus, it unveils a hidden universality of this structure. We also give a general relation between the orthogonal polynomials related to the determinantal structure and the skew-orthogonal polynomials corresponding to the Pfaffian. As a particular example, we consider the chiral unitary ensembles in great detail.

At low temperature the low end of the QCD Dirac spectrum is well described by chiral random matrix theory. In contrast, at high temperature there is no similar statistical description of the spectrum. We show that at high temperature the lowest part of the spectrum consists of a band of statistically uncorrelated eigenvalues obeying essentially Poisson statistics and the corresponding eigenvectors are extremely localized. Going up in the spectrum the spectral density rapidly increases and the eigenvectors become more and more delocalized. At the same time the spectral statistics gradually crosses over to the bulk statistics expected from the corresponding random matrix ensemble. This phenomenon is reminiscent of Anderson localization in disordered conductors. Our findings are based on staggered Dirac spectra in quenched lattice simulations with the SU(2) gauge group.

We generalize the recently discovered relationship between JT gravity and double-scaled random matrix theory to the case that the boundary theory may have time-reversal symmetry and may have fermions with or without supersymmetry. The matching between variants of JT gravity and matrix ensembles depends on the assumed symmetries. Time-reversal symmetry in the boundary theory means that unorientable spacetimes must be considered in the bulk. In such a case, the partition function of JT gravity is still related to the volume of the moduli space of conformal structures, but this volume has a quantum correction and has to be computed using Reidemeister-Ray-Singer "torsion." Presence of fermions in the boundary theory (and thus a symmetry $(-1)^F$) means that the bulk has a spin or pin structure. Supersymmetry in the boundary means that the bulk theory is associated to JT supergravity and is related to the volume of the moduli space of super Riemann surfaces rather than of ordinary Riemann surfaces. In all cases we match JT gravity or supergravity with an appropriate random matrix ensemble. All ten standard random matrix ensembles make an appearance -- the three Dyson ensembles and the seven Altland-Zirnbauer ensembles. To facilitate the analysis, we extend to the other ensembles techniques that are most familiar in the case of the original Wigner-Dyson ensemble of hermitian matrices. We also generalize Mirzakhani's recursion for the volumes of ordinary moduli space to the case of super Riemann surfaces.

The unitary Wilson random matrix theory is an interpolation between the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble. This new way of interpolation is also reflected in the orthogonal polynomials corresponding to such a random matrix ensemble. Although the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble are associated with the Dyson index β = 2, the intermediate ensembles exhibit a mixing of orthogonal polynomials and skew-orthogonal polynomials. We consider the Hermitian and the non-Hermitian Wilson random matrix and derive the corresponding polynomials, their recursion relations, Christoffel–Darboux-like formulas, Rodrigues formulas and representations as random matrix averages in a unifying way. With the help of these results, we derive the unquenched k-point correlation function of the Hermitian and the non-Hermitian Wilson random matrix in terms of two-flavor partition functions only. This representation is due to a Pfaffian factorization. It drastically simplifies the expressions, which can be easily numerically evaluated. It also serves as a good starting point for studying the Wilson–Dirac operator in the ϵ-regime of lattice quantum chromodynamics.

We consider random normal matrix and planar symplectic ensembles, which can be interpreted as two-dimensional Coulomb gases having determinantal and Pfaffian structures, respectively. For a class of radially symmetric potentials with soft edges, we derive the asymptotic expansions of the log-partition functions up to and including the O(1)-terms as the number N of particles increases. Notably, our findings stress that the formulas of the $$O(\log N)$$ - and O(1)-terms in these expansions depend on the connectivity of the droplet. For random normal matrix ensembles, our formulas agree with the predictions proposed by Zabrodin and Wiegmann up to an additive constant depending on N but not on the background potential. For planar symplectic ensembles, the expansions contain a new kind of ingredient in the O(N)-terms, the logarithmic potential evaluated at the origin in addition to the entropy of the ensembles.

There is considerable interest in understanding the relation between random-matrix ensembles and quantum chaotic systems in the context of the universality of energy-level correlations. In this connection, while Gaussian ensembles of random matrices have been studied extensively, not much is known about ensembles with non-Gaussian weight functions. Dyson has shown that the n-level correlation functions can be expressed in terms of a kernel function involving orthogonal and skew-orthogonal polynomials--orthogonal for matrix ensembles with unitary invariance and skew orthogonal for ensembles with orthogonal and symplectic invariances. We have obtained the following results. (1) Skew-orthogonal polynomials of both types are derived for the Jacobi class of weight functions including the limiting cases of associated Laguerre and Hermite (or Gaussian). (2) Matrix-integral representations are given for the general weight functions. (3) Asymptotic forms of the polynomials are obtained rigorously for the Jacobi class and in the form of an ansatz for the general case. (4) For the three types of ensembles, the (asymptotic) n-level correlation functions with appropriate scaling are shown to be universal, being independent of the weight function and location in the spectrum, and identical with the well-known Gaussian results. This provides a rigorous justification for the universality of the Gaussian ensemble results observed in quantum chaotic systems. As expected, the level density is not universal.

We study the spectral properties of and spectral crossovers between different random matrix ensembles [Poissonian, Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE)] in correlated spin-chain systems, in the presence of random magnetic fields, and the scalar spin-chirality term, competing with the usual isotropic and time-reversal invariant Heisenberg term. We have investigated these crossovers in the context of the level-spacing distribution and the level-spacing ratio distribution. We use random matrix theory (RMT) analytical results to fit the observed Poissonian-to-GOE and GOE-to-GUE crossovers, and examine the relationship between the RMT crossover parameter $\ensuremath{\lambda}$ and scaled physical parameters of the spin-chain systems in terms of a scaling exponent. We find that the crossover behavior exhibits universality, in the sense that, it becomes independent of lattice size in the large Hamiltonian matrix dimension limit. Moreover, the scaling exponent obtained from such a finite size scaling analysis, seems to be quite robust and independent of the type of crossover considered or the specific spectral correlation measure used.

This chapter is devoted to statistical nuclear physics with SU(3) in the context of statistical nuclear spectroscopy and random matrix theory. First, some general aspects of trace propagation are described and trace propagation formulas for traces over SU(3) irreps, in SM and IBM spaces, in four different situations are given with applications to test the goodness of SU(3) symmetry in (2s1d) shell nuclei and also for understanding regularities generated by random interactions. Going beyond this, presented also are : (i) results for nuclear level density enhancement in deformed nuclei using the SU(3) symmetry of sdIBM-1; (ii) brief derivation of the asymptotic formula for the SU(3) multiplicities in the \(\{m\} \rightarrow (\lambda \mu )\) reduction for \(U((\eta +1)(\eta +2)/2) \supset SU(3)\); (iii) some results for the EGUE(2)-SU(3) random matrix ensemble with SU(3) symmetry.

We develop a perturbative approach to study the supersymmetric nonlinear sigma model characterized by a generic coupling matrix in the strong coupling limit. The method allows us to calculate explicitly the moments of the eigenfunctions and the two-level correlation function in the lowest order of the perturbative expansion. We find that the obtained expressions are equivalent to the results derived before for the corresponding random matrix ensembles. Such an equivalence is elucidated and generalized to all orders of the perturbative expansion by mapping the sigma model onto the field theory describing the almost diagonal random matrices.

We calculate the time-evolution of a discrete-time fragmentation process in which clusters of particles break up and reassemble and move stochastically with size-dependent rates. In the continuous-time limit the process turns into the totally asymmetric simple exclusion process (only pieces of size 1 break off a given cluster). We express the exact solution of master equation for the process in terms of a determinant which can be derived using the Bethe ansatz. From this determinant we compute the distribution of the current across an arbitrary bond which after appropriate scaling is given by the distribution of the largest eigenvalue of the Gaussian unitary ensemble of random matrices. This result confirms universality of the scaling form of the current distribution in the KPZ universality class and suggests that there is a link between integrable particle systems and random matrix ensembles.

This paper focuses on large neural networks whose synaptic connectivity matrices are randomly chosen from certain random matrix ensembles. The dynamics of these networks can be characterized by the eigenvalue spectra of their connectivity matrices. In reality, neurons in a network do not necessarily behave in a similar way, but may belong to several different categories. The first study of the spectra of two-component neural networks was carried out by Rajan and Abbott [Phys. Rev. Lett. 97, 188104 (2006)]. In their model, neurons are either "excitatory" or "inhibitory," and strengths of synapses from different types of neurons have Gaussian distributions with different means and variances. A surprising finding by Rajan and Abbott is that the eigenvalue spectra of these types of random synaptic matrices do not depend on the mean values of their elements. In this paper we prove that this is true even for a much more general type of random neural network, where there is a finite number of types of neurons and their synaptic strengths have correlated distributions. Furthermore, using the diagrammatic techniques, we calculate the explicit formula for the spectra of synaptic matrices of multicomponent neural networks.

We give expansions of reproducing kernels of the Christoffel–Darboux type in terms of Schur polynomials. For this, we use evaluations of averages of characteristic polynomials and Schur polynomials in random matrix ensembles. We explicitly compute new Schur averages, such as the Schur average in a q-Laguerre ensemble, and the ensuing expansions of random matrix kernels. In addition to classical and q-deformed cases on the real line, we use extensions of Dotsenko–Fateev integrals to obtain expressions for kernels on the complex plane. Moreover, a known interplay between Wronskians of Laguerre polynomials, Painlevé tau functions and conformal block expansions is discussed in relationship to the Schur expansion obtained.

We study CMV matrices (discrete one-dimensional Dirac-type operators) with random decaying coefficients. Under mild assumptions, we identify the local eigenvalue statistics in the natural scaling limit. For rapidly decreasing coefficients, the eigenvalues have rigid spacing (like the numerals on a clock); in the case of slow decrease, the eigenvalues are distributed according to a Poisson process. For a certain critical rate of decay, we obtain the β-ensembles of random matrix theory. The temperature β−1 appears as the square of the coupling constant

Reinforcement learning models have the potential to clarify meaningful individual differences in the decision-making process. This study focused on two aspects regarding the nature of a reinforcement learning model and its parameters: the problems of model misspecification and reliability. Online participants, N = 453, completed self-report measures and a probabilistic learning task twice 1.5 months apart, and data from the task were fitted using several reinforcement learning models. To address the problem of model misspecification, we compared the models with and without the influence of choice history, or perseveration. Results showed that the lack of a perseveration term in the model led to a decrease in learning rates for win and loss outcomes, with slightly different influences depending on outcome volatility, and increases in inverse temperature. We also conducted simulations to examine the mechanism of the observed biases and revealed that failure to incorporate perseveration directly affected the estimation bias in the learning rate and indirectly affected that in inverse temperature. Furthermore, in both model fittings and model simulations, the lack of perseveration caused win-stay probability underestimation and loss-shift probability overestimation. We also assessed the parameter reliability. Test–retest reliabilities were poor (learning rates) to moderate (inverse temperature and perseveration magnitude). A learning effect was noted in the inverse temperature and perseveration magnitude parameters, showing an increment of the estimates in the second session. We discuss possible misinterpretations of results and limitations considering the estimation biases and parameter reliability.

We study the large $N$ limit of a sparse random block matrix ensemble. It depends on two parameters: the average connectivity $Z$ and the size of the blocks $d$, which is the dimension of an euclidean space. In the limit of large $d$, with $\frac{Z}{d}$ fixed, we prove the conjecture that the spectral distribution of the sparse random block matrix converges in the case of the Adjacency block matrix to the one of the effective medium approximation, in the case of the Laplacian block matrix to the Marchenko-Pastur distribution. We extend previous analytical computations of the moments of the spectral density of the Adjacency block matrix and the Lagrangian block matrix, valid for all values of $Z$ and $d$.