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Register a new userEikonal formulation of large dynamical random matrix models
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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.
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We consider a general Langevin dynamics for the one-dimensional N-particle Coulomb gas with confining potential $V$ at temperature $beta$. These dynamics describe for $beta=2$ the time evolution of the eigenvalues of $Ntimes N$ random Hermitian matrices. The equilibrium partition function -- equal to the normalization constant of the Laughlin wave function in fractional quantum Hall effect -- is known to satisfy an infinite number of constraints called Virasoro or loop constraints. We introduce here a dynamical generating function on the space of random trajectories which satisfies a large class of constraints of geometric origin. We focus in this article on a subclass induced by the invariance under the Schrodinger-Virasoro algebra.
In this paper we consider a particular version of the random walk with restarts: random reset events which bring suddenly the system to the starting value. We analyze its relevant statistical properties like the transition probability and show how an equilibrium state appears. Formulas for the first-passage time, high-water marks and other extreme statistics are also derived: we consider counting problems associated naturally to the system. Finally we indicate feasible generalizations useful for interpreting different physical effects.
The purpose of this article is to study the eigenvalues $u_1^{, t}=e^{ittheta_1},dots,u_N^{,t}=e^{ittheta_N}$ of $U^t$ where $U$ is a large $Ntimes N$ random unitary matrix and $t>0$. In particular we are interested in the typical times $t$ for which all the eigenvalues are simultaneously close to $1$ in different ways thus corresponding to recurrence times in the issue of quantum measurements. Our strategy consists in rewriting the problem as a random matrix integral and use loop equations techniques to compute the first orders of the large $N$ asymptotic. We also connect the problem to the computation of a large Toeplitz determinant whose symbol is the characteristic function of several arc segments of the unit circle. In particular in the case of a single arc segment we recover Widoms formula. Eventually we explain why the first return time is expected to converge towards an exponential distribution when $N$ is large. Numeric simulations are provided along the paper to illustrate the results.
We consider the model of random sequential adsorption, with depositing objects, as well as those already at the surface, decreasing in size according to a specified time dependence, from a larger initial value to a finite value in the large time limit. Numerical Monte Carlo simulations of two-dimensional deposition of disks and one-dimensional deposition of segments are reported for the density-density correlation function and gap-size distribution function, respectively. Analytical considerations supplement numerical results in the one-dimensional case. We investigate the correlation hole - the depletion of correlation functions near contact and, for the present model, their vanishing at contact - that opens up at finite times, as well as its closing and reemergence of the logarithmic divergence of correlation properties at contact in the large time limit.
We collect and systematize general definitions and facts on the application of quantum groups to the construction of functional relations in the theory of integrable systems. As an example, we reconsider the case of the quantum group $U_q(mathcal L(mathfrak{sl}_2))$ related to the six-vertex model. We prove the full set of the functional relations in the form independent of the representation of the quantum group in the quantum space and specialize them to the case of the six-vertex model.
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