Related papers: Approximate Ultrametricity for Random Measures and…
We prove that the Hilbert space description of all joint von Neumann measurements on a quantum state can be reproduced in terms of a single measure space ({\Omega}, F, {\mu}) with a normalized real-valued measure {\mu}, that is, in terms of…
Spherical spin glasses are canonical models for smooth random functions in high dimensions. In this review, we survey several interrelated lines of research on their geometric structure. We begin with results concerning critical points and…
Within the wide class of disordered materials, spin glasses occupy a special place because of their conceptually simple definition of randomly interacting spins. Their modelling has triggered spectacular developments of out-of-equilibrium…
We present a new approach to study measures on ensembles of contours, polymers or other objects interacting by some sort of exclusion condition. For concreteness we develop it here for the case of Peierls contours. Unlike existing methods,…
We develop a general framework for the probabilistic analysis of random finite point clouds in the context of topological data analysis. We extend the notion of a barcode of a finite point cloud to compact metric spaces. Such a barcode…
We study the problem of estimating, in the sense of optimal transport metrics, a measure which is assumed supported on a manifold embedded in a Hilbert space. By establishing a precise connection between optimal transport metrics, optimal…
The distribution function of particles over clusters is proposed for a system of identical intersecting spheres, the centres of which are uniformly distributed in space. Consideration is based on the concept of the rank number of clusters,…
Solomonoff's central result on induction is that the posterior of a universal semimeasure M converges rapidly and with probability 1 to the true sequence generating posterior mu, if the latter is computable. Hence, M is eligible as a…
The aim of this paper is to extend the concept of measure density introduced by Buck for finite unions of arithmetic progressions, to arbitrary subsets of N defined by a given system of decompositions. This leads to a variety of new…
We investigate the dynamics of spin glasses from the `rheological' point of view, in which aging is suppressed by the action of small, non-conservative forces. The different features can be expressed in terms of the scaling of relaxation…
We consider the problem of algorithmically sampling from the Gibbs measure of a mixed $p$-spin spherical spin glass. We give a polynomial-time algorithm that samples from the Gibbs measure up to vanishing total variation error, for any…
Accurate approximation of probability measures is essential in numerical applications. This paper explores the quantization of probability measures using the maximum mean discrepancy (MMD) distance as a guiding metric. We first investigate…
Random matrix ensembles are introduced that respect the local tensor structure of Hamiltonians describing a chain of $n$ distinguishable spin-half particles with nearest-neighbour interactions. We prove a central limit theorem for the…
This is a short review about recent methods and results, mostly for mean field spin glasses, based on interpolation and comparison schemes. In particular, the Parisi spontaneous replica symmetry breaking phenomenon is described in the frame…
We study the problem of fitting an ultrametric distance to a dissimilarity graph in the context of hierarchical cluster analysis. Standard hierarchical clustering methods are specified procedurally, rather than in terms of the cost function…
The mean field spin glass model is analyzed by a combination of mathematically rigororous methods and a powerful Ansatz. The method exploited is general, and can be applied to others disordered mean field models such as, e.g., neural…
We introduce the notion of uniform exactness, or uniform amenability at infinity, for discrete groups and prove it for a wide class of groups containing free groups and their limit groups. This shows a novel strong convergence phenomenon…
We study the fluctuation problems at high temperature in the general mixed $p$-spin glass models under the weak external field assumption: $h= \rho N^{-\alpha}, \rho>0, \alpha \in [1/4,\infty]$. By extending the cluster expansion approach…
Consider a set P of N random points on the unit sphere of dimension $d-1$, and the symmetrized set S = P union (-P). The halving polyhedron of S is defined as the convex hull of the set of centroids of N distinct points in S. We prove that…
We consider diffraction at random point scatterers on general discrete point sets in $\R^\nu$, restricted to a finite volume. We allow for random amplitudes and random dislocations of the scatterers. We investigate the speed of convergence…