120 research outputs found
Multi-class oscillating systems of interacting neurons
We consider multi-class systems of interacting nonlinear Hawkes processes
modeling several large families of neurons and study their mean field limits.
As the total number of neurons goes to infinity we prove that the evolution
within each class can be described by a nonlinear limit differential equation
driven by a Poisson random measure, and state associated central limit
theorems. We study situations in which the limit system exhibits oscillatory
behavior, and relate the results to certain piecewise deterministic Markov
processes and their diffusion approximations.Comment: 6 figure
Summaries
Приведены рефераты статей данного номера на английском языке
Summaries
Приведены рефераты статей данного номера на английском языке
The scaling limit of the critical one-dimensional random Schrodinger operator
We consider two models of one-dimensional discrete random Schrodinger
operators (H_n \psi)_l ={\psi}_{l-1}+{\psi}_{l +1}+v_l {\psi}_l,
{\psi}_0={\psi}_{n+1}=0 in the cases v_k=\sigma {\omega}_k/\sqrt{n} and
v_k=\sigma {\omega}_k/ \sqrt{k}. Here {\omega}_k are independent random
variables with mean 0 and variance 1.
We show that the eigenvectors are delocalized and the transfer matrix
evolution has a scaling limit given by a stochastic differential equation. In
both cases, eigenvalues near a fixed bulk energy E have a point process limit.
We give bounds on the eigenvalue repulsion, large gap probability, identify the
limiting intensity and provide a central limit theorem.
In the second model, the limiting processes are the same as the point
processes obtained as the bulk scaling limits of the beta-ensembles of random
matrix theory. In the first model, the eigenvalue repulsion is much stronger.Comment: 36 pages, 2 figure
SLE and Virasoro representations: localization
We consider some probabilistic and analytic realizations of Virasoro
highest-weight representations. Specifically, we consider measures on paths
connecting points marked on the boundary of a (bordered) Riemann surface. These
Schramm-Loewner Evolution (SLE)- type measures are constructed by the method of
localization in path space. Their partition function (total mass) is the
highest-weight vector of a Virasoro representation, and the action is given by
Virasoro uniformization.
We review the formalism of Virasoro uniformization, which allows to define a
canonical action of Virasoro generators on functions (or sections) on a -
suitably extended - Teichm\"uller space. Then we describe the construction of
families of measures on paths indexed by marked bordered Riemann surfaces.
Finally we relate these two notions by showing that the partition functions of
the latter generate a highest-weight representation - the quotient of a
reducible Verma module - for the former.Comment: 59 pages. To appear in Comm. Math. Phy
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