Related papers: Matrix Models and Large-N Behavior

For a sufficiently nice 2 dimensional shape, we define its approximating matrix (or patterned matrix) as a random matrix with iid entries arranged according to a given pattern. For large approximating matrices, we observe that the…

I provide evidence that the 2D $RP^{N-1}$ model for $N \ge 3$ is equivalent to the $O(N)$-invariant non-linear $\sigma$-model in the continuum limit. To this end, I mainly study particular versions of the models, to be called constraint…

Recently, a convergent series employing a non-Gaussian initial approximation was constructed and shown to be an effective computational tool for the finite size lattice models with a polynomial interaction. Here we numerically examine the…

The $J$-matrix method is extended to difference and $q$-difference operators and is applied to several explicit differential, difference, $q$-difference and second order Askey-Wilson type operators. The spectrum and the spectral measures…

We conjecture that the $O(N)$-symmetric non-linear sigma model in the semi-infinite $(1+1)$-dimensional space is ``integrable'' with respect to the ``free'' and the ``fixed'' boundary conditions. We then derive, for both cases, the boundary…

This is a significantly expanded version of the survey paper "Mixing and decay of correlations in non-uniformly expanding maps: a survey of recent results" math/0301319. We discuss recent results on decay of correlations for non-uniformly…

In this paper, we propose a general mathematical framework to represent many multi-agent signalling systems in recent works. Our goal is to apply previous results in monotonicity to this class of systems and study their asymptotic behavior.…

We consider the Landau-Ginzburg-Wilson Hamiltonian with O(n)x O(m) symmetry and compute the critical exponents at all fixed points to O(n^{-2}) and to O(\epsilon^3) in a \epsilon=4-d expansion. We also consider the corresponding non-linear…

We propose a method for characterizing large complex networks by introducing a new matrix structure, unique for a given network, which encodes structural information; provides useful visualization, even for very large networks; and allows…

The renormalization group approach is studied for large $N$ models. The approach of Br\'ezin and Zinn-Justin is explained and examined for matrix models. The validity of the approach is clarified by using the vector model as a similar and…

Inverse problems are in many cases solved with optimization techniques. When the underlying model is linear, first-order gradient methods are usually sufficient. With nonlinear models, due to nonconvexity, one must often resort to…

Matrix generalizations of the N-particle quantum-mechanical Calogero model classifying according to representations of the symmetric group $S_N$ are considered. Symmetry properties of the eigen-wave functions in the matrix Calogero models…

We present a concise survey of matrix completion methods and associated implementations of several fundamental algorithms. Our study covers both passive and adaptive strategies. We further illustrate the behavior of a simple adaptive…

We explore new symmetries in two-component third-order Burgers' type systems in (1+1)-dimension using Wang's O-scheme. We also find a master symmetry for a (2+1)-dimensional Davey-Stewartson type system. These results shed light on the…

We present our progress on a study of the $O(3)$ model in two-dimensions using the Tensor Renormalization Group method. We first construct the theory in terms of tensors, and show how to construct $n$-point correlation functions. We then…

This paper advances the study of multivariate function approximation using neural network operators activated by symmetrized and perturbed hyperbolic tangent functions. We propose new non-linear operators that preserve dynamic symmetries…

The algebraic formulation of Large N matrix mechanics recently developed by Halpern and Schwartz leads to a practical method of numerical computation for both action and Hamiltonian problems. The new technique posits a boundary condition on…

The orthogonal group acts on the space of several $n\times n$ matrices by simultaneous conjugation. For an infinite field of characteristic different from two, relations between generators for the algebra of invariants are described. As an…

We introduce a family of maps generating continued fractions where the digit $1$ in the numerator is replaced cyclically by some given non-negative integers $(N_1,\ldots,N_m)$. We prove the convergence of the given algorithm, and study the…

Neuroscience has recently made much progress, expanding the complexity of both neural-activity measurements and brain-computational models. However, we lack robust methods for connecting theory and experiment by evaluating our new big…