## A. Marcus On Equivalences between Blocks of Group Algebras: Reduction to the Simple Components

Reference: LMENS-95-1 (December 94)
Source: dvi, ps

Abstract: A conjecture of Michel Broué states that if D is an abelian Sylow p-subgroup of a finite group G, and H = NG (D), then the principal blocks of G and H are Rickard equivalent. The structure of groups with abelian Sylow p-subgroups, as determined by P. Fong and M.E. Harris, raises the following question: assuming that Broué's conjecture holds for the simple components of G, under what conditions does it hold for G itself? Due to the structure of G, this problem requires mainly the lifting of Rickard complexes to p0-extensions of the simple components and the construction of complexes over wreath products. We give here these reduction steps, which may be regarded as a "Clifford theory" of tilting complexes.

AMS Classification: Primary 20C05; Secondary 20C20
Keywords: blocks withh abelian defect groups, Morita equivalences, derived equivalences, p-permutation modules

## G. Ben Arous, F. Castell A Probabilistic Approach to Semi-Classical Approximations

Reference: LMENS-95-2 (February 95)
Source: dvi, ps

Abstract:We study in this paper the semi-classical expansion of the Schrödinger equation, using a probabilistic approach based on the Wiener measure. Using almost-analytic extensions, we exhibit a probabilistic ansatz for the wave function. We show that this ansatz approximates very well the wave function in the semi-classical regime, and gives the semi-classical expansion under mild hypothesis on the potential at infinity, and no analyticity conditions. In this paper, the study takes place before the caustics.

AMS Classification: 35C15, 60H10, 60H30, 81Q20

## G. Ben Arous, A. Guionnet Large Deviations for Langevin Spin Glass Dynamics

Reference: LMENS-95-3 (February 95)
Source: dvi, ps

Abstract: We study the asymptotic behavior of asymmetrical spin glass dynamics in a Sherrington-Kirkpatrick model as proposed by Sompolinsky-Zippelius. We prove that the annealed law of the empirical measure on path space of these dynamics satisfy a large deviation principle in the high temperature regime. We study the rate function of this large deviation principle and prove that it achieves its minimum value at a unique probability measure Q which is not markovian.We deduce that the quenched law of the empirical measure converges to $\delta_{Q}$. Extending then the preceeding results to replicated dynamics, we investigate the quenched behavior of a single spin. We get quenched convergence to Q in the case of a symmetric initial law and even potential for the free spin.

AMS Classification: 60F10, 60H10, 60K35, 82C44, 82C31, 82C22
Keywords: Large deviations, Interacting random processes, Statistical mechanics, Langevin dynamics

## X. Chen, D. Hilhorst, E. Logak Asymptotic Behavior of Solutions of an Allen-Cahn Equation with a Non-local Term

Reference: LMENS-95-4 (March 95)
Source: dvi, ps

Abstract: We consider an Allen-Cahn equation involving a spatial average term, which arises as a singular limit of a reaction-diffusion system modelling phase transition. We show that the solution of this equation converges to a step function taking values 1 on both sides of a moving interface. The normal velocity of the interface is given by the sum of mean curvature and of some non-local term depending on the volume delimited by the interface. We prove existence and uniqueness of a smooth solution of this free-boundary problem locally in time.

Keywords: Allen-Cahn equation, Integro-differential equations, Singular limit, Propagation of interfaces

## G. Ben Arous, A. Guionnet Annealed and Quenched Propagation of Chaos for Asymmetric Spin Glass Dynamics

Reference: LMENS-95-5 (March 95)
Source: dvi, ps

Abstract: We study the asymptotic behavior of asymmetrical spin glass dynamics in a Sherrington-Kirkpatrick model as proposed by Sompolinsky-Zippelius. We prove, without any condition on time and temperature, an annealed propagation of chaos result. Extending this result to replicated systems, we conclude that the law of a single spin converges to a non markovian probability measure, in law with respect to the random interaction.

AMS Classification: 60F10, 60H10, 60K35, 82C44, 82C31, 82C22
Keywords: Large deviations, Interacting random processes, Statistical mechanics, Langevin dynamics.

## E. Grenier Oscillatory Perturbations of the Navier Stokes Equations

Reference: LMENS-95-6 (March 95)
Source: dvi, ps

Abstract: In this paper, we study the convergence of weak and strong solutions of oscillatory perturbations of the Navier-Stokes equations and in particular the asymptotic behaviour of rotating fluids and of slightly compressible fluids.

AMS Classification: 35B25, 35C20, 35Q30, 76U05
Keywords: Fluid mechanics, Singular perturbations, Asymptotic expansions

## B. Helffer, A. Mohamed Semiclassical Analysis for the Ground State Energy of a Schr\"oding Operator with Magnetic Wells

Reference: LMENS-95-7 (April 95)
Source: dvi, ps

Abstract: Motivated by a recent paper by Montgomery [?], we give the asymptotic behavior, in the semi-classical sense, of the ground state energy for the Schr"odinger operator with a magnetic field. We consider the case when the locus of the minima of the intensity of the magnetic field is compact and our study is sharper when this locus is an hypersurface or a finite union of points.

## A. Beauville Quantum cohomology of complete intersection

Reference: LMENS-95-8 (March 95)
Source: dvi, ps

Abstract: The quantum cohomology algebra of a projective manifold X is the cohomology H(X,Q) endowed with a different algebra structure, which takes into account the geometry of rational curves in X. We show that this algebra takes a remarkably simple form for complete intersections when the dimension is large enough w.r.t. the degree. As a reward we get a number of surprising enumerative formulas relating lines, conics and twisted cubics on X.

## A. Beauville A Calabi-Yau threefold with non-Abelian fundamental group

Reference: LMENS-DIVERS-95-1 (Février 95)
Source: dvi

Abstract: On construit un exemple d'une variété de Calabi-Yau de dimension 3 dont le groupe fondamental est le groupe quaternionien à 8 éléments.

## A. Trouvé An approach of pattern recognition though infinite dimensional group action

Reference: LMENS-95-9 (May 95)
Source: dvi, ps

Abstract: Non rigid deformations of patterns can be interpreted as the action of an infinite dimensional group A(n) on a given set P of patterns. Following Lie group ideas, a small deformation can be well described by an element y of the tangent space at identity TeA(n). Given a metric n on TeA(n), which brings the cost of a small deformation, we show that we can define on A(n) a left invariant distance dA(n)which gives the distance between two arbitrary large deformations.
This allows to reformulate in a unified framework many pattern recognition tasks. Finally, we propose a sub-optimal algorithm to solve three important classes of pattern recognition problems through a gradient algorithm on A(n) whose convergence is rigorously established.

Keywords: Deformable modele, Infinite dimensional Lie group, Geodesic distance, Pattern recognition

## Y. Brenier A homogenized model for vortex sheets

Reference: LMENS-95-10 (May 95)
Source: dvi, ps

## C. Bolley, B. Helffer Proof of the De Gennes formula for the superheating field in the weak k limit

Reference: LMENS-95-11 (June 95)
Source: dvi, ps

Abstract: In continuation with our preceding paper [3] concerning the superconducting film, we present in this article new estimates for the superheating field in the weak k limit. The principal result is the proof of the existence of a finite superheating field h sh,+(k) (obtained by restricting the usual definition of the superheating field to solutions of the Ginzburg-Landau system (f,A) with f positive) in the case of a semi-infinite interval. The bound is optimal in the limit k -> 0 and permits to prove (combining with our previous results) the De Gennes formula
2[-3/4]= lim k->0 k[1/2] h[sh,+] (k)
The proof is obtained by improving slightly the estimates given in [3] where an upper bound was found but under the additional condition that the function f was bounded from below by some fixed constant ae > 0.

## R. Goldin, B. Teissier Resolving singularities of plane analytic branches with one toric morphism

Reference: LMENS-95-12 (June 95)

## B. Teissier Valuative sorites

Reference: LMENS-95-12 (June 95)

## O. Catoni A new upper-bound for the free energy of the Sherrington-Kirkpatrick model of a spin glass

Reference: LMENS-95-13 (June 95)
Source: ps

AMS Classification: Statistical Mechanics

## B. Helffer On Laplace integrals and transfer operators in large dimension: examples in the non convex case

Reference: LMENS-95-14 (July 95)
Source: dvi, ps

Abstract: This paper gives detailed proofs concerning the analysis of the pair correlations for a non convex model. Using the transfer matrix approach, the problem is reduced to the analysis of the spectral properties of this transfer operators. Although the problem is similar to the semiclassical study of the Kac's operator presented in our paper with M.Brunaud [BruHe91] which was devoted 1to the study of exp-v/2.exph2 .exp-v/2for h small, new features appear for the model exp-v/2h.exph .exp-v/2h. Our principal results concern the splitting between the two largest eigenvalues of this operator. We give an uper bound and a lower bound for this splitting in the semi-classical regim. As a corollary, we get a good control of the decay of the pair corre lation. Some of the results were announced in [He94e]. WKB related constructions will be developed in [He95c] (see the announcement in [He95b]).

## M. Vergne, M. Brion Lattice points in simple polytopes

Reference: LMENS-95-15 (September 95)
Source: dvi, ps

## F. Sauvageot Principe de densité pour les groupes réductifs

Reference: LMENS-95-16 (Septembre 95)

## A. Trouvé Diffeomorphisms groups and pattern mathching in image analysis

Reference: LMENS-95-17 (September 95)
Source: dvi, ps

Abstract: In a previous paper, the author proposes to see the deformations of a common pattern as the action of an infinite dimensional group. We show in this paper that this approach can be applied numerically for pattern matching in image analysis of digital images. Using Lie group ideas, we construct a distance between deformations defined through a metric given the cost of infinitesimal deformations. Then we propose a numerical scheme to solve a variational problem involving this distance and leading to a sub-optimal pattern matching.

## F. Merle, H. Zaag Stability of Blow-up profile for equation of the type $u_t=\Delta u+|u|^{p-1}u$

Reference: LMENS-95-18 (October 95)
Source: dvi, ps

Abstract: In this paper, we consider the following nonlinear equation
$u_t=\Delta u+|u|^{p-1}u$
u(.,)= u_0,
(and various extensions of this equation, where the maximum principle do not apply). We first describe precisely the behavior of a blow-up solution near blow-up time and point. We then show a stability result on this be- havior.

(AMS)Classification: 35K, 35B35, 35B40
Keywords: Blow-up, Profile, Stability

## G. Ben Arous, M. Gradinaru Singularities of hypoelliptic Green functions

Reference: LMENS-95-19 (October 95)
Source: dvi, ps

Abstract: This paper is devoted to a precise description of the singularity near the diagonal of the Green function associated to a hypoelliptic operator using a probabilistic approach. Examples and some applications to potential theory are given.

(AMS)Classification 1991 : 60J60, 35H05, 60H10, 60J45
Keywords: Hypoelliptic operator, Green function, degenerate diffusion, Taylor stochastic expansion, capacity.

## W. Werner Bounds for disconnection exponents

Reference: LMENS-95-20 (November 95)
Source: dvi, ps

Abstract: We slightly inprove the upper bounds of disconnection exponents for planar Brownian motion that we derived in an earlier paper. We also give a plain proof of the lower bound 1/(2\pi) for the disconnection exponent for one path.

## E.E. Puckette, W. Werner Monte-Carlo tests and conjectures for disconnection exponents

Reference: LMENS-95-21 (November 95)
Source: dvi, ps

Abstract: Using Monte-Carlo simulations, we estimate numerically disconnection exponents for planar Brownian motions. These simulations tend to confirm conjectures by Duplantier and Mandelbrot.

## J. Bertoin, W. Werner Stable windings

Reference: LMENS-95-22 (November 95)
Source: dvi, ps

Abstract: We derive the asymptotic laws of winding numbers for planar isotropic stable Lévy processes and walks of index $\alpha\in(0,2)$.

## O. Catoni, R. Cerf The exit path of a Markov chain with rare transitions

Reference: LMENS-95-23 (November 95)
Abstract: We study the exit path from a general domain after the last visit to a set of a Markov chain with rare transitions. We prove several large deviation principles for the law of the succession of the cycles visited by the process (the cycle path), the succession of the saddle points gone through to jump from cycle to cycle on the cycle path (the saddle path) and the succession of all the points gone through (the exit path). We estimate the time the process spends in each cycle of the cycle path and how it decomposes into the time spent in each point of the exit path. We describe a systematic method to find the most likely saddle paths. We apply these results to the reversible case of the Metropolis dynamics.

## E. Leichtnam, F. Golse, M. StenzelIntrinsic microlocal analysis and inversion formulae for the heat equation on compact real-analytic riemannian manifolds

Reference: LMENS-95-24 (November 95)
Source: dvi, ps

Abstract: This paper is devoted to a new intrinsic description of microlocal analytic singularities on a connected compact C\omega Riemannian manifold (X,g). In this approach, the microlocal singularities of a distribution u on X are described in terms of the growth, as t -> 0+, of the analytic extension of e-t\delta u to a suitable complexification X' of X, identified with a tubular neighborhood of the zero section in T*X. First we show that the analytic extension of the heat kernel of (X,g) to X' is an F.B.I. transform in the sense of Sj"ostrand. Then we establish various inversion formulae for the heat semigroup e-t\delta

## G. Ben Arous, R. Cerf Metastability of the Three Dimensional Ising Model on a Torus at very low Temperatures

Reference: LMENS-95-25 (November 95)
Source: ps

Abstract: We study the metastability of the stochastic three dimensional Ising model on a finite torus under a small positive magnetic field at very low temperatures

## G. Ben Arous, J.D. Deuschel The construction of the d+1-dimensional gaussian droplet

Reference: LMENS-95-26 (November 95)
Source: dvi, ps

Abstract: The aim of this note is to study the asymptotic behavior of a gaussian random field, under the condition that the variables are positive and the total volume under the variables converges to some fixed number v > 0. In the context of Statistical Mechanics, this corresponds to the problem of constructing a droplet on a hard wall with a given volume. We show that, properly rescaled, the profile of a gaussian configuration converges to a smooth hypersurface, which solves a quadratic variational problem. Our main tool is a scaling dependent large deviation principle for random hypersurfaces.

## L. Alonso, R. Cerf The three dimensional polyominoes of minimal area

Source: tar

Abstract: The set of the three dimensional polyominoes of minimal area and of volume ~$n$ contains a polyomino which is the union of a quasicube $j\times (j+\delta)\times (j+\theta)$, $\delta,\theta\in\{0,1\}$, a quasisquare $l\times (l+\epsilon)$, $\epsilon\in\{0,1\}$, and a bar $k$. This shape is naturally associated to the unique decomposition of ~$n=j(j+\delta)(j+\theta)+l(l+\epsilon)+k$ as the sum of a maximal quasicube, a maximal quasisquare and a bar. For ~$n$ a quasicube plus a quasisquare, or a quasicube minus one, the minimal polyominoes are reduced to these shapes. The minimal area is explicitly computed and yields a discrete isoperimetric inequality. These variational problems are the key for finding the path of escape from the metastable state for the three dimensional Ising model at very low temperatures

(AMS)Classification:05B50 51M25 82C44

## H.S. Dumas, L. Dumas, F. Golse On the mean free path for a periodic array of spherical obstacles

Reference: LMENS-95-27 (November 95)
Source: dvi, ps

Abstract: We prove theorems pertaining to periodic arrays of spherical obstacles which show how the macroscopic limit of the mean free path depends on the scaling of the size of the obstacles. We treat separately the cases where the obstacles are totally and partially absorbing, and we also distinguish between two-dimensional arrays, where our results are optimal, and higher dimensional arrays where they are not. The cubically symmetric arrays to which these results apply do not have finite horizon.

Keywords: Lorentz gas, kinetic theory, mean free path, continued fractions, ergodization rate, small divisors.

## G. Ben Arous, A. Guionnet Symmetric langevin spin glass dynamics

Reference: LMENS-95-28 (November 95)

## C. Bardos, L. Dumas, F. Golse Diffusion approximation for billards with totally accomodating scatterers

Reference: LMENS-95-29 (November 95)
Source: ps

Abstract: This article is aimed at studying the 2D motion of independent point particles colliding with a periodic array of circular obstacles. The interaction between the particles and the obstacles is described by a total accomodation reflection law. Assuming that the array of scatterers has finite horizon, the density of particles is approximated by the solution of a diffusion equation in the long time and large scale regime. The proof relies on a multi-scale asymptotics and gives the order of approximation. The reader is referred to [BDG] where the present work was announced.

## Y. Brenier, L. Corrias A Kinetic formulation for multi-branch entropy solutions of scalar conservation laws

Reference: LMENS-95-30 (december 95)
Abstract: Multivalued solutions with a limited number of branches of the inviscid Burgers equation can be obtained by solving closed systems of moment equations. For thos purpose, a suitable concept of entropy multivaluated solutions with K branches is introduced.

## M. Brion, M. Vergne An equivariant Riemann-Roch theorem for complete, simplicial toric varieties

Reference: LMENS-95-31 (december 95)

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