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In this work, we examine spectral properties of Markov transition operators corresponding to Gaussian perturbations of discrete time dynamical systems on the circle. We develop a method for calculating asymptotic expressions for eigenvalues (in the zero noise limit) and show that changes to the number or period of stable orbits for the deterministic system correspond to changes in the number of limiting modulus 1 eigenvalues of the transition operator for the perturbed process. We call this phenomenon a $lambda$-bifurcation. Asymptotic expressions for the corresponding eigenfunctions and eigenmeasures are also derived and are related to Hermite functions.
We explicitly determine the spectrum of transfer operators (acting on spaces of holomorphic functions) associated to analytic expanding circle maps arising from finite Blaschke products. This is achieved by deriving a convenient natural representation of the respective adjoint operators.
The random convex hull of a Poisson point process in $mathbb{R}^d$ whose intensity measure is a multiple of the standard Gaussian measure on $mathbb{R}^d$ is investigated. The purpose of this paper is to invent a new viewpoint on these Gaussian polyt
In this paper, we consider a strongly-repelling model of $n$ ordered particles ${e^{i theta_j}}_{j=0}^{n-1}$ with the density $p({theta_0},cdots, theta_{n-1})=frac{1}{Z_n} exp left{-frac{beta}{2}sum_{j eq k} sin^{-2} left( frac{theta_j-theta_k}{2}ri
In this paper we analyze a simple spectral method (EIG1) for the problem of matrix alignment, consisting in aligning their leading eigenvectors: given two matrices $A$ and $B$, we compute $v_1$ and $v_1$ two corresponding leading eigenvectors. The al
We explore the boundaries of sine kernel universality for the eigenvalues of Gaussian perturbations of large deterministic Hermitian matrices. Equivalently, we study for deterministic initial data the time after which Dysons Brownian motion exhibits