31 research outputs found

### Entropies, convexity, and functional inequalities

Our aim is to provide a short and self contained synthesis which generalise
and unify various related and unrelated works involving what we call
Phi-Sobolev functional inequalities. Such inequalities related to Phi-entropies
can be seen in particular as an inclusive interpolation between Poincare and
Gross logarithmic Sobolev inequalities. In addition to the known material,
extensions are provided and improvements are given for some aspects. Stability
by tensor products, convolution, and bounded perturbations are addressed. We
show that under simple convexity assumptions on Phi, such inequalities hold in
a lot of situations, including hyper-contractive diffusions, uniformly strictly
log-concave measures, Wiener measure (paths space of Brownian Motion on
Riemannian Manifolds) and generic Poisson space (includes paths space of some
pure jumps Levy processes and related infinitely divisible laws). Proofs are
simple and relies essentially on convexity. We end up by a short parallel
inspired by the analogy with Boltzmann-Shannon entropy appearing in Kinetic
Gases and Information Theories.Comment: Formerly "On Phi-entropies and Phi-Sobolev inequalities". Author's
www homepage: http://www.lsp.ups-tlse.fr/Chafai

### Circular law for non-central random matrices

Let $(X_{jk})_{j,k\geq 1}$ be an infinite array of i.i.d. complex random
variables, with mean 0 and variance 1. Let \la_{n,1},...,\la_{n,n} be the
eigenvalues of $(\frac{1}{\sqrt{n}}X_{jk})_{1\leq j,k\leq n}$. The strong
circular law theorem states that with probability one, the empirical spectral
distribution \frac{1}{n}(\de_{\la_{n,1}}+...+\de_{\la_{n,n}}) converges
weakly as $n\to\infty$ to the uniform law over the unit disc
\{z\in\dC;|z|\leq1\}. In this short note, we provide an elementary argument
that allows to add a deterministic matrix $M$ to $(X_{jk})_{1\leq j,k\leq n}$
provided that $\mathrm{Tr}(MM^*)=O(n^2)$ and \mathrm{rank}(M)=O(n^\al) with
\al<1. Conveniently, the argument is similar to the one used for the
non-central version of Wigner's and Marchenko-Pastur theorems.Comment: accepted in Journal of Theoretical Probabilit

### Glauber versus Kawasaki for spectral gap and logarithmic Sobolev inequalities of some unbounded conservative spin systems

Inspired by the recent results of C. Landim, G. Panizo and H.-T. Yau [LPY] on
spectral gap and logarithmic Sobolev inequalities for unbounded conservative
spin systems, we study uniform bounds in these inequalities for Glauber
dynamics of Hamiltonian of the form V(x_1) + ... + V(x_n) + V(M-x_1 -...-x_n),
(x_1,...,x_n) in R^n Specifically, we examine the case V is strictly convex (or
small perturbation of strictly convex) and, following [LPY], the case V is a
bounded perturbation of a quadratic potential. By a simple path counting
argument for the standard random walk, uniform bounds for the Glauber dynamics
yields, in a transparent way, the classical L^{-2} decay for the Kawasaki
dynamics on d-dimensional cubes of length L. The arguments of proofs however
closely follow and make heavy use of the conservative approach and estimates of
[LPY], relying in particular on the Lu-Yau martingale decomposition and clever
partitionings of the conditional measure.Comment: 20 pages. Accepted for publication in ``Markov Processes and Related
Fields'

### Around the circular law

These expository notes are centered around the circular law theorem, which
states that the empirical spectral distribution of a nxn random matrix with
i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the
complex plane as the dimension $n$ tends to infinity. This phenomenon is the
non-Hermitian counterpart of the semi circular limit for Wigner random
Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random
covariance matrices. We present a proof in a Gaussian case, due to Silverstein,
based on a formula by Ginibre, and a proof of the universal case by revisiting
the approach of Tao and Vu, based on the Hermitization of Girko, the
logarithmic potential, and the control of the small singular values. Beyond the
finite variance model, we also consider the case where the entries have heavy
tails, by using the objective method of Aldous and Steele borrowed from
randomized combinatorial optimization. The limiting law is then no longer the
circular law and is related to the Poisson weighted infinite tree. We provide a
weak control of the smallest singular value under weak assumptions, using
asymptotic geometric analysis tools. We also develop a quaternionic
Cauchy-Stieltjes transform borrowed from the Physics literature.Comment: Added: one reference and few comment

### Explicit formulas for a continuous stochastic maturation model. Application to anticancer drug pharmacokinetics/pharmacodynamics

We present a continuous time model of maturation and survival, obtained as
the limit of a compartmental evolution model when the number of compartments
tends to infinity. We establish in particular an explicit formula for the law
of the system output under inhomogeneous killing and when the input follows a
time-inhomogeneous Poisson process. This approach allows the discussion of
identifiability issues which are of difficult access for finite compartmental
models. The article ends up with an example of application for anticancer drug
pharmacokinetics/pharmacodynamics.Comment: Revised version, accepted for publication in Stochastic Models
(Taylor & Francis

### Dynamics of a planar Coulomb gas

We study the long-time behavior of the dynamics of interacting planar
Brow-nian particles, confined by an external field and subject to a singular
pair repulsion. The invariant law is an exchangeable Boltzmann -- Gibbs
measure. For a special inverse temperature, it matches the Coulomb gas known as
the complex Ginibre ensemble. The difficulty comes from the interaction which
is not convex, in contrast with the case of one-dimensional log-gases
associated with the Dyson Brownian Motion. Despite the fact that the invariant
law is neither product nor log-concave, we show that the system is well-posed
for any inverse temperature and that Poincar{\'e} inequalities are available.
Moreover the second moment dynamics turns out to be a nice Cox -- Ingersoll --
Ross process in which the dependency over the number of particles leads to
identify two natural regimes related to the behavior of the noise and the speed
of the dynamics.Comment: Minor revision for Annals of Applied Probabilit

### Spectrum of large random reversible Markov chains: two examples

We take on a Random Matrix theory viewpoint to study the spectrum of certain
reversible Markov chains in random environment. As the number of states tends
to infinity, we consider the global behavior of the spectrum, and the local
behavior at the edge, including the so called spectral gap. Results are
obtained for two simple models with distinct limiting features. The first model
is built on the complete graph while the second is a birth-and-death dynamics.
Both models give rise to random matrices with non independent entries.Comment: accepted in ALEA, March 201

### Concentration for Coulomb gases and Coulomb transport inequalities

We study the non-asymptotic behavior of Coulomb gases in dimension two and
more. Such gases are modeled by an exchangeable Boltzmann-Gibbs measure with a
singular two-body interaction. We obtain concentration of measure inequalities
for the empirical distribution of such gases around their equilibrium measure,
with respect to bounded Lipschitz and Wasserstein distances. This implies
macroscopic as well as mesoscopic convergence in such distances. In particular,
we improve the concentration inequalities known for the empirical spectral
distribution of Ginibre random matrices. Our approach is remarkably simple and
bypasses the use of renormalized energy. It crucially relies on new
inequalities between probability metrics, including Coulomb transport
inequalities which can be of independent interest. Our work is inspired by the
one of Ma{\"i}da and Maurel-Segala, itself inspired by large deviations
techniques. Our approach allows to recover, extend, and simplify previous
results by Rougerie and Serfaty.Comment: Improvement on an assumption, and minor modification