Sampling the Fermi statistics and other conditional product measures

Through a Metropolis-like algorithm with single step computational cost of order one, we build a Markov chain that relaxes to the canonical Fermi statistics for k non-interacting particles among m energy levels. Uniformly over the temperature as well as the energy values and degeneracies of the energy levels we give an explicit upper bound with leading term km(ln k) for the mixing time of the dynamics. We obtain such construction and upper bound as a special case of a general result on (non-homogeneous) products of ultra log-concave measures (like binomial or Poisson laws) with a global constraint. As a consequence of this general result we also obtain a disorder-independent upper bound on the mixing time of a simple exclusion process on the complete graph with site disorder. This general result is based on an elementary coupling argument and extended to (non-homogeneous) products of log-concave measures.Comment: 21 page

On soft capacities, quasi-stationary distributions and the pathwise approach to metastability

Motivated by the study of the metastable stochastic Ising model at subcritical temperature and in the limit of a vanishing magnetic field, we extend the notion of ($\kappa$, $\lambda$)-capacities between sets, as well as the associated notion of soft-measures, to the case of overlapping sets. We recover their essential properties, sometimes in a stronger form or in a simpler way, relying on weaker hypotheses. These properties allow to write the main quantities associated with reversible metastable dynamics, e.g. asymptotic transition and relaxation times, in terms of objects that are associated with two-sided variational principles. We also clarify the connection with the classical "pathwise approach" by referring to temporal means on the appropriate time scale.Comment: 29 pages, 1 figur

Recurrence and transience for long-range reversible random walks on a random point process

We consider reversible random walks in random environment obtained from symmetric long--range jump rates on a random point process. We prove almost sure transience and recurrence results under suitable assumptions on the point process and the jump rate function. For recurrent models we obtain almost sure estimates on effective resistances in finite boxes. For transient models we construct explicit fluxes with finite energy on the associated electrical network.Comment: 34 page

Random Forests and Networks Analysis

D. Wilson~\cite{[Wi]} in the 1990's described a simple and efficient algorithm based on loop-erased random walks to sample uniform spanning trees and more generally weighted trees or forests spanning a given graph. This algorithm provides a powerful tool in analyzing structures on networks and along this line of thinking, in recent works~\cite{AG1,AG2,ACGM1,ACGM2} we focused on applications of spanning rooted forests on finite graphs. The resulting main conclusions are reviewed in this paper by collecting related theorems, algorithms, heuristics and numerical experiments. A first foundational part on determinantal structures and efficient sampling procedures is followed by four main applications: 1) a random-walk-based notion of well-distributed points in a graph 2) how to describe metastable dynamics in finite settings by means of Markov intertwining dualities 3) coarse graining schemes for networks and associated processes 4) wavelets-like pyramidal algorithms for graph signals.Comment: Survey pape

High Speed Blanking: An Experimental Method to Measure Induced Cutting Forces

Lien vers la version éditeur: http://link.springer.com/article/10.1007/s11340-013-9738-1A new blanking process that involves punch speed up to 10 ms −1 has obvious advantages in increased productivity. However, the inherent dynamics of such a process makes it difficult to develop a practical high speed punch press. The fracture phenomenon governing the blanking process has to be well understood to correctly design the machine support and the tooling. To observe this phenomenon at various controlled blanking speeds a specific experimental device has been developed. The goal is to measure accurately the shear blanking forces imposed on the specimen during blanking. In this paper a new method allowing the blanking forces to be measured and taking into account the proposed test configuration is explained. This technique has been used to determine the blanking forces experienced when forming C40 steel and quantifies the effect of process parameters such as punch die clearance, punch speed, and sheet metal thickness on the blanking force evolution

A Dirichlet principle for non reversible Markov chains and some recurrence theorems

27 pagesInternational audienceWe extend the Dirichlet principle to non-reversible Markov processes on countable state spaces. We present two variational formulas for the solution of the Poisson equation or, equivalently, for the capacity between two disjoint sets. As an application we prove some recurrence theorems. In particular, we show the recurrence of two-dimensional cycle random walks under a second moment condition on the winding numbers

Loop-erased partitioning of a graph: mean-field analysis

We consider a random partition of the vertex set of an arbitrary graph that can be sampled using loop-erased random walks stopped at a random independent exponential time of parameter $q>0$, that we see as a tuning parameter.The related random blocks tend to cluster nodes visited by the random walk on time scale $1/q$. We explore the emerging macroscopic structure by analyzing 2-point correlations. To this aim, it is defined an interaction potential between pair of vertices, as the probability that they do not belong to the same block of the random partition. This interaction potential can be seen as an affinity measure for ``densely connected nodes'' and capture well-separated regions in network models presenting non-homogeneous landscapes. In this spirit, we compute this potential and its scaling limits on a complete graph and on a non-homogeneous weighted version with community structures. For the latter geometry we show a phase-transition for ``community detectability'' as a function of the tuning parameter and the edge weights.Comment: 30 pages, 1 figur

Estimating the inverse trace using random forests on graphs

International audienc