119,634 research outputs found
The cluster index of regularly varying sequences with applications to limit theory for functions of multivariate Markov chains
We introduce the cluster index of a multivariate regularly varying stationary
sequence and characterize the index in terms of the spectral tail process. This
index plays a major role in limit theory for partial sums of regularly varying
sequences. We illustrate the use of the cluster index by characterizing
infinite variance stable limit distributions and precise large deviation
results for sums of multivariate functions acting on a stationary Markov chain
under a drift condition
The Fermi-Pasta-Ulam problem: 50 years of progress
A brief review of the Fermi-Pasta-Ulam (FPU) paradox is given, together with
its suggested resolutions and its relation to other physical problems. We focus
on the ideas and concepts that have become the core of modern nonlinear
mechanics, in their historical perspective. Starting from the first numerical
results of FPU, both theoretical and numerical findings are discussed in close
connection with the problems of ergodicity, integrability, chaos and stability
of motion. New directions related to the Bose-Einstein condensation and quantum
systems of interacting Bose-particles are also considered.Comment: 48 pages, no figures, corrected and accepted for publicatio
Chain of matrices, loop equations and topological recursion
Random matrices are used in fields as different as the study of
multi-orthogonal polynomials or the enumeration of discrete surfaces. Both of
them are based on the study of a matrix integral. However, this term can be
confusing since the definition of a matrix integral in these two applications
is not the same. These two definitions, perturbative and non-perturbative, are
discussed in this chapter as well as their relation. The so-called loop
equations satisfied by integrals over random matrices coupled in chain is
discussed as well as their recursive solution in the perturbative case when the
matrices are Hermitean.Comment: 28 pages, 1 figure, contribution to The Oxford Handbook of Random
Matrix Theor
Finite-size and correlation-induced effects in Mean-field Dynamics
The brain's activity is characterized by the interaction of a very large
number of neurons that are strongly affected by noise. However, signals often
arise at macroscopic scales integrating the effect of many neurons into a
reliable pattern of activity. In order to study such large neuronal assemblies,
one is often led to derive mean-field limits summarizing the effect of the
interaction of a large number of neurons into an effective signal. Classical
mean-field approaches consider the evolution of a deterministic variable, the
mean activity, thus neglecting the stochastic nature of neural behavior. In
this article, we build upon two recent approaches that include correlations and
higher order moments in mean-field equations, and study how these stochastic
effects influence the solutions of the mean-field equations, both in the limit
of an infinite number of neurons and for large yet finite networks. We
introduce a new model, the infinite model, which arises from both equations by
a rescaling of the variables and, which is invertible for finite-size networks,
and hence, provides equivalent equations to those previously derived models.
The study of this model allows us to understand qualitative behavior of such
large-scale networks. We show that, though the solutions of the deterministic
mean-field equation constitute uncorrelated solutions of the new mean-field
equations, the stability properties of limit cycles are modified by the
presence of correlations, and additional non-trivial behaviors including
periodic orbits appear when there were none in the mean field. The origin of
all these behaviors is then explored in finite-size networks where interesting
mesoscopic scale effects appear. This study leads us to show that the
infinite-size system appears as a singular limit of the network equations, and
for any finite network, the system will differ from the infinite system
Algebraic methods in random matrices and enumerative geometry
We review the method of symplectic invariants recently introduced to solve
matrix models loop equations, and further extended beyond the context of matrix
models. For any given spectral curve, one defined a sequence of differential
forms, and a sequence of complex numbers Fg . We recall the definition of the
invariants Fg, and we explain their main properties, in particular symplectic
invariance, integrability, modularity,... Then, we give several example of
applications, in particular matrix models, enumeration of discrete surfaces
(maps), algebraic geometry and topological strings, non-intersecting brownian
motions,...Comment: review article, Latex, 139 pages, many figure
Precise large deviations for dependent regularly varying sequences
We study a precise large deviation principle for a stationary regularly
varying sequence of random variables. This principle extends the classical
results of A.V. Nagaev (1969) and S.V. Nagaev (1979) for iid regularly varying
sequences. The proof uses an idea of Jakubowski (1993,1997) in the context of
centra limit theorems with infinite variance stable limits. We illustrate the
principle for \sv\ models, functions of a Markov chain satisfying a polynomial
drift condition and solutions of linear and non-linear stochastic recurrence
equations
Elementary submodels in infinite combinatorics
The usage of elementary submodels is a simple but powerful method to prove
theorems, or to simplify proofs in infinite combinatorics. First we introduce
all the necessary concepts of logic, then we prove classical theorems using
elementary submodels. We also present a new proof of Nash-Williams's theorem on
cycle-decomposition of graphs, and finally we improve a decomposition theorem
of Laviolette concerning bond-faithful decompositions of graphs
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