Six-vertex model and random matrix distributions

The ice model and the eight-vertex model on the two-dimensional Sierpinski gasket

A connection between integrability properties and general statistical properties of the spectra of symmetric transfer matrices of the asymmetric eight-vertex model is studied using random matrix theory (eigenvalue spacing distribution and spectral rigidity). For Yang-Baxter integrable cases, including free-fermion solutions, we have found a Poissonian behavior, whereas level repulsion close to the Wigner distribution is found for non-integrable models. For the asymmetric eight-vertex model, however, the level repulsion can also disappearand the Poisson distribution be recovered on (non Yang--Baxter integrable) algebraic varieties, the so-called disorder varieties. We also present an infinite set of algebraic varieties which are stable under the action of an infinite discrete symmetry group of the parameter space. These varieties are possible loci for free parafermions. Using our numerical criterion we have tested the generic calculability of the model on these algebraic varieties.

A quantum relativistic integrable model as the continuous limit of the six-vertex model

Mobile communication applications provide a unique data source for the research of human mobility pattern. Based on the distribution data of urban mobile phone users, in this paper is explored the macroscopic dynamical behavior of urban mobility human by using the method of random matrix theory. The largest eigenvalue and the corresponding eigenvector of mobile phone user data deviate far from the distribution of random matrix. The deviations from random matrix vary with time. We find that the largest eigenvalue corresponds to a whole behavior common to all urban human mobility. The results indicate the temporal trends of the mean of correlation coefficient and the largest eigenvalue. We also find that the spatio temporal evolution of the weight of eigenvector components for the eigenvector corresponding to the largest eigenvalue is very consistent with the fluctuation pattern of the macroscopic behavior of urban human mobility.

We study a new integrable probabilistic system, defined in terms of a stochastic colored vertex model on a square lattice. The main distinctive feature of our model is a new family of parameters attached to diagonals rather than to rows or columns, like in other similar models. Because of these new parameters the previously known results about vertex models cannot be directly applied, but nevertheless the integrability remains, and we prove explicit integral expressions for q-deformed moments of the (colored) height functions of the model. Following known techniques our model can be interpreted as a q-discretization of the Beta polymer model from (Probab Theory Relat Fields 167(3):1057–1116 (2017). arXiv:1503.04117) with a new family of parameters, also attached to diagonals. To demonstrate how integrability with respect to the new diagonal parameters works, we extend the known results about Tracy–Widom large-scale fluctuations of the Beta polymer model.

Solutions of the reflection equation for the [formula omitted] vertex model

These lecture notes provide a comprehensive self-contained introduction to the analysis of Wishart matrix moments. This study may act as an introduction to some particular aspects of random matrix theory or as a self-contained exposition of Wishart matrix moments. Random matrix theory plays a central role in statistical physics computational mathematics and engineering sciences including data assimilation signal processing combinatorial optimization compressed sensing econometrics and mathematical finance among numerous others. The mathematical foundations of the theory of random matrices lies at the intersection of combinatorics non-commutative algebra geometry multivariate functional and spectral analysis and of course statistics and probability theory. As a result most of the classical topics in random matrix theory are technical and mathematically difficult to penetrate for non-experts and regular users and practitioners. The technical aim of these notes is to review and extend some important results in random matrix theory in the specificcontext of real random Wishart matrices. This special class of Gaussian-type sample covariance matrix plays an important role in multivariate analysis and in statistical theory. We derive non-asymptotic formulae for the full matrix moments of real valued Wishart random matrices. As a corollary we derive and extend a number of spectral and trace-type results for the case of non-isotropic Wishart random matrices. We also derive the full matrix moment analogues of some classic spectral and trace-type moment results. For example we derive semi-circle and Marchencko-Pastur-type laws in the non-isotropic and full matrix cases. Laplace matrix transforms and matrix moment estimates are also studied along with new spectral and trace concentration-type inequalities.

We develop a new method for studying the asymptotics of symmetric polynomials of representation-theoretic origin as the number of variables tends to infinity. Several applications of our method are presented: We prove a number of theorems concerning characters of infinite-dimensional unitary group and their $q$-deformations. We study the behavior of uniformly random lozenge tilings of large polygonal domains and find the GUE-eigenvalues distribution in the limit. We also investigate similar behavior for alternating sign matrices (equivalently, six-vertex model with domain wall boundary conditions). Finally, we compute the asymptotic expansion of certain observables in $O(n=1)$ dense loop model.

The thermal disordering of the $d$-density wave, proposed to be the origin of the pseudogap state of high temperature superconductors, is suggested to be the same as that of the statistical mechanical model known as the 6-vertex model. The low temperature phase consists of a staggered order parameter of circulating currents, while the disordered high temperature phase is a power-law phase with no order. A special feature of this transition is the complete lack of an observable specific heat anomaly at the transition. There is also a transition at a even higher temperature at which the magnitude of the order parameter collapses. These results are due to classical thermal fluctuations and are entirely unrelated to a quantum critical point in the ground state. The quantum mechanical ground state can be explored by incorporating processes that causes transitions between the vertices, allowing us to discuss quantum phase transition in the ground state as well as the effect of quantum criticality at a finite temperature as distinct from the power-law fluctuations in the classical regime. A generalization of the model on a triangular lattice that leads to a 20-vertex model may shed light on the Wigner glass picture of the metal-insulator transition in two-dimensional electron gas. The power-law ordered high temperature phase may be generic to a class of constrained systems and its relation to recent advances in the quantum dimer models is noted.

Solvable vertex models in two dimensions are of importance in conformal field theory, phase transitions and integrable models. We consider here the Dn spin vertex models, for n which is odd. The models involve also the anti–spinor representation. We describe here the Boltzmann weights for these representations using crossing symmetry from the previously known spinor representations. For calculation reasons we limit ourself to the n=3 and n=5 cases, which are described explicitly.

The bus system in Cuernavaca, Mexico and its connections to random matrix distributions have been the subject of an interesting recent study by M Krbálek and P Seba in [15, 16]. In this paper we introduce and analyse a microscopic model for the bus system. We show that introducing a natural repulsion does produce random matrix distributions in natural double scaling regimes. The techniques employed include non-intersecting paths, logarithmic potential theory, determinantal point processes, and asymptotic analysis of several orthogonal polynomial ensembles. In addition, we introduce a circular bus model and include various calculations of non-crossing probabilities.

The free-fermion point refers to a GL ( 2 ) × GL ( 1 ) parametrized Yang-Baxter equation within the six-vertex model. It has been known for a long time that this is connected with the quantum group U q ( gl ( 1 | 1 ) ) . We demonstrate that R -matrices from the finite quantum superalgebra U q ( gl ( 1 | 1 ) ) produce a dense subset of the free-fermionic Yang-Baxter equations of the six-vertex model, matching those of the prime, simple modules in the affine quantum superalgebra U q ( gl ˆ ( 1 | 1 ) ) . Either of these quantum groups can be used to generate the full free-fermion point, and we discuss them both. Our discussion includes 6 families of six-vertex models used by Brubaker, Bump, and Friedberg in connection with Tokuyama's theorem, a deformation of the Weyl character formula. Thus our work gives quantum group interpretations for those models, known informally as Tokuyama ice.

The quantum inverse scattering method is a scheme for solving integrable models in dimensions, building on an R-matrix that satisfies the Yang–Baxter equation (YBE) and in terms of which one constructs a commuting family of transfer matrices. In the standard formulation, this R-matrix acts on a tensor product of vector spaces. Here, we relax this tensorial property and develop a framework for describing and analysing integrable models based on planar algebras, allowing non-separable R-operators satisfying generalised YBEs. We also re-evaluate the notion of integrals of motion and characterise when an (algebraic) transfer operator is polynomial in a single integral of motion. We refer to such models as polynomially integrable. In an eight-vertex model, we demonstrate that the corresponding transfer operator is polynomial in the natural Hamiltonian. In the Temperley–Lieb loop model with loop fugacity , we likewise find that, for all but finitely many β-values, the transfer operator is polynomial in the usual Hamiltonian element of the Temperley–Lieb algebra , at least for . Moreover, we find that this model admits a second canonical Hamiltonian, and that this Hamiltonian also acts as a polynomial integrability generator for small n and all but finitely many β-values.

Introduction to algorithms for Monte Carlo simulations and their application to QCD

In this paper we study the usefulness of the algebraic approach to the quantum field theory in nonflat backgrounds. The correspondances between some models of statistical mechanics and quantum field theory in curved space-times are also discussed.