The Rescaled Pólya Urn and the Wright–Fisher Process with Mutation

In recent papers the authors introduce, study and apply a variant of the Eggenberger–Pólya urn, called the “rescaled” Pólya urn, which, for a suitable choice of the model parameters, exhibits a reinforcement mechanism mainly based on the last observations, a random persistent fluctuation of the predictive mean and the almost sure convergence of the empirical mean to a deterministic limit. In this work, motivated by some empirical evidence, we show that the multidimensional Wright–Fisher diffusion with mutation can be obtained as a suitable limit of the predictive means associated to a family of rescaled Pólya urns.

Asymptotic Phase and Amplitude for Classical and Semiclassical Stochastic Oscillators via Koopman Operator Theory

The asymptotic phase is a fundamental quantity for the analysis of deterministic limit-cycle oscillators, and generalized definitions of the asymptotic phase for stochastic oscillators have also been proposed. In this article, we show that the asymptotic phase and also amplitude can be defined for classical and semiclassical stochastic oscillators in a natural and unified manner by using the eigenfunctions of the Koopman operator of the system. We show that the proposed definition gives appropriate values of the phase and amplitude for strongly stochastic limit-cycle oscillators, excitable systems undergoing noise-induced oscillations, and also for quantum limit-cycle oscillators in the semiclassical regime.

Random Perturbations of Matrix Polynomials

A sum of a large-dimensional random matrix polynomial and a fixed low-rank matrix polynomial is considered. The main assumption is that the resolvent of the random polynomial converges to some deterministic limit. A formula for the limit of the resolvent of the sum is derived, and the eigenvalues are localised. Four instances are considered: a low-rank matrix perturbed by the Wigner matrix, a product HX of a fixed diagonal matrix H and the Wigner matrix X and two special matrix polynomials of higher degree. The results are illustrated with various examples and numerical simulations.

Rectangular Young tableaux and the Jacobi ensemble

International audience It has been shown by Pittel and Romik that the random surface associated with a large rectangular Youngtableau converges to a deterministic limit. We study the fluctuations from this limit along the edges of the rectangle.We show that in the corner, these fluctuations are gaussian whereas, away from the corner and when the rectangle isa square, the fluctuations are given by the Tracy-Widom distribution. Our method is based on a connection with theJacobi ensemble.

Transport coherence and diffusion in a temporal asymmetric rocked ratchet model

We study the overdamped Brownian dynamics of a particle in a sawtooth potential along with a temporal asymmetric driving force. We observe that in the deterministic limit, the transport coherence which is determined by a dimensionless quantity, called Peclet number Pe, is quite high under certain circumstances. For all the regime of parameter space of this model, Pe in our model shows similar features of current like Stokes efficiency. Diffusion as a function of driving amplitude shows a nonmonotonic behavior and results a minimum exactly at which the current shows a maximum. Unlike the previously studied models, Pe in our model shows a peaking behavior with temperature. Moreover, the diffusion shows a nonlinear dependence of temperature in the long-time limit and it is sensitive to the potential asymmetry parameter.

Discrete derivative asymptotics of the β-Hermite eigenvalues

We consider the asymptotics of the difference between the empirical measures of the β-Hermite tridiagonal matrix and its minor. We prove that this difference has a deterministic limit and Gaussian fluctuations. Through a correspondence between measures and continual Young diagrams, this deterministic limit is identified with the Vershik–Kerov–Logan–Shepp curve. Moreover, the Gaussian fluctuations are identified with a sectional derivative of the Gaussian free field.