Matrix models for classical groups and Toeplitz±\pm Hankel minors with applications to Chern-Simons theory and fermionic models

We study matrix integration over the classical Lie groups $U(N),Sp(2N),O(2N)$ and $O(2N+1)$, using symmetric function theory and the equivalent formulation in terms of determinants and minors of Toeplitz$\pm$Hankel matrices. We establish a number of factorizations and expansions for such integrals, also with insertions of irreducible characters. As a specific example, we compute both at finite and large $N$ the partition functions, Wilson loops and Hopf links of Chern-Simons theory on $S^{3}$ with the aforementioned symmetry groups. The identities found for the general models translate in this context to relations between observables of the theory. Finally, we use character expansions to evaluate averages in random matrix ensembles of Chern-Simons type, describing the spectra of solvable fermionic models with matrix degrees of freedom.Comment: 32 pages, v2: Several improvements, including a Conclusions and Outlook section, added. 36 page

Toeplitz minors and specializations of skew Schur polynomials

We express minors of Toeplitz matrices of finite and large dimension in terms of symmetric functions. Comparing the resulting expressions with the inverses of some Toeplitz matrices, we obtain explicit formulas for a Selberg-Morris integral and for specializations of certain skew Schur polynomials.Comment: v2: Added new results on specializations of skew Schur polynomials, abstract and title modified accordingly and references added; v3: final, published version; 18 page

Memory effects can make the transmission capability of a communication channel uncomputable

Most communication channels are subjected to noise. One of the goals of Information Theory is to add redundancy in the transmission of information so that the information is transmitted reliably and the amount of information transmitted through the channel is as large as possible. The maximum rate at which reliable transmission is possible is called the capacity. If the channel does not keep memory of its past, the capacity is given by a simple optimization problem and can be efficiently computed. The situation of channels with memory is less clear. Here we show that for channels with memory the capacity cannot be computed to within precision 1/5. Our result holds even if we consider one of the simplest families of such channels -information-stable finite state machine channels-, restrict the input and output of the channel to 4 and 1 bit respectively and allow 6 bits of memory.Comment: Improved presentation and clarified claim

A large deviation principle for empirical measures on Polish spaces: Application to singular Gibbs measures on manifolds

We prove a large deviation principle for a sequence of point processes defined by Gibbs probability measures on a Polish space. This is obtained as a consequence of a more general Laplace principle for the non-normalized Gibbs measures. We consider three main applications: Conditional Gibbs measures on compact spaces, Coulomb gases on compact Riemannian manifolds and the usual Gibbs measures in the Euclidean space. Finally, we study the generalization of Fekete points and prove a deterministic version of the Laplace principle known as $\Gamma$-convergence. The approach is partly inspired by the works of Dupuis and co-authors. It is remarkably natural and general compared to the usual strategies for singular Gibbs measures.Comment: 23 pages, abstract also in french, for more details in the proofs see version 1, application to Gaussian polynomials adde

Edge fluctuations for random normal matrix ensembles

A famous result going back to Eric Kostlan states that the moduli of the eigenvalues of random normal matrices with radial potential are independent yet non identically distributed. This phenomenon is at the heart of the asymptotic analysis of the edge, and leads in particular to the Gumbel fluctuation of the spectral radius when the potential is quadratic. In the present work, we show that a wide variety of laws of fluctuation are possible, beyond the already known cases, including for instance Gumbel and exponential laws at unusual speeds. We study the convergence in law of the spectral radius as well as the limiting point process at the edge. Our work can also be seen as the asymptotic analysis of the edge of two-dimensional determinantal Coulomb gases and the identification of the limiting kernels.Comment: 43 pages, improved version with more general theorem

Chern-Simons theory encoded on a spin chain

We construct a 1d spin chain Hamiltonian with generic interactions and prove that the thermal correlation functions of the model admit an explicit random matrix representation. As an application of the result, we show how the observables of $U(N)$ Chern-Simons theory on $S^{3}$ can be reproduced with the thermal correlation functions of the 1d spin chain, which is of the XX type, with a suitable choice of exponentially decaying interactions between infinitely many neighbours. We show that for this model, the correlation functions of the spin chain at a finite temperature $\beta =1$ give the Chern-Simons partition function, quantum dimensions and the full topological $S$-matrix.Comment: v2, 11 pages. Expanded, more detailed version. Misprints correcte