pinecone/core-2020-05-10-deduplication huggingface dataset:
"Zero'' temperature stochastic 3D Ising model and dimer covering fluctuations: a first step towards interface mean curvature motion

International audienceWe consider the Glauber dynamics for the Ising model with "+" boundary conditions, at zero temperature or at a temperature that goes to zero with the system size (hence the quotation marks in the title). In dimension d = 3 we prove that an initial domain of linear size L of "−" spins disappears within a time τ+, which is at most L2(log L)c and at least L2/(c log L) for some c > 0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the motion by mean curvature that is expected to describe, on large time scales, the evolution of the interface between "+" and "−" domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimmer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factors, is the first rigorous confirmation of the Lifshitz law τ+ ≃ const × L2, conjectured on heuristic grounds [8, 13]. In dimension d = 2, τ+ can be shown to be of order L2 without logarithmic corrections: the upper bound was proven in [6], and here we provide the lower bound. For d = 2, we also prove that the spectral gap of the generator behaves like equation image for L large, as conjectured in [2]
A new frequency-uniform coercive boundary integral equation for acoustic scattering

A new boundary integral operator is introduced for the solution of the soundsoft acoustic scattering problem, i.e., for the exterior problem for the Helmholtz equation with Dirichlet boundary conditions. We prove that this integral operator is coercive in L2(Γ) (where Γ is the surface of the scatterer) for all Lipschitz star-shaped domains. Moreover, the coercivity is uniform in the wavenumber k = ω/c, where ω is the frequency and c is the speed of sound. The new boundary integral operator, which we call the “star-combined” potential operator, is a slight modification of the standard combined potential operator, and is shown to be as easy to implement as the standard one. Additionally, to the authors' knowledge, it is the only second-kind integral operator for which convergence of the Galerkin method in L2(Γ) is proved without smoothness assumptions on Γ except that it is Lipschitz. The coercivity of the star-combined operator implies frequency-explicit error bounds for the Galerkin method for any approximation space. In particular, these error estimates apply to several hybrid asymptoticnumerical methods developed recently that provide robust approximations in the high-frequency case. The proof of coercivity of the star-combined operator critically relies on an identity first introduced by Morawetz and Ludwig in 1968, supplemented further by more recent harmonic analysis techniques for Lipschitz domains. © 2011 Wiley Periodicals, Inc
Optimal Large-Time Behavior of the Vlasov-Maxwell-Boltzmann System in
  the Whole Space

In this paper we study the large-time behavior of classical solutions to the
two-species Vlasov-Maxwell-Boltzmann system in the whole space $\R^3$. The
existence of global in time nearby Maxwellian solutions is known from [34] in
2006. However the asymptotic behavior of these solutions has been a challenging
open problem. Building on our previous work [10] on time decay for the simpler
Vlasov-Poisson-Boltzmann system, we prove that these solutions converge to the
global Maxwellian with the optimal decay rate of $O(t^{-3/2+\frac{3}{2r}})$ in
$L^2_\xi(L^r_x)$-norm for any $2\leq r\leq \infty$ if initial perturbation is
smooth enough and decays in space-velocity fast enough at infinity. Moreover,
some explicit rates for the electromagnetic field tending to zero are also
provided.Comment: 42 pages, updated the file according to the many helpful and valuable
  comments of the refere
On the fragmentation of a torus by random walk

We consider a simple random walk on a discrete torus (Z/NZ)^d with dimension
d at least 3 and large side length N. For a fixed constant u > 0, we study the
percolative properties of the vacant set, consisting of the set of vertices not
visited by the random walk in its first [uN^d] steps. We prove the existence of
two distinct phases of the vacant set in the following sense: if u > 0 is
chosen large enough, all components of the vacant set contain no more than a
power of log(N) vertices with high probability as N tends to infinity. On the
other hand, for small u > 0, there exists a macroscopic component of the vacant
set occupying a non degenerate fraction of the total volume N^d. In dimensions
d at least 5, we additionally prove that this macroscopic component is unique,
by showing that all other components have volumes of order at most a power of
log(N). Our results thus solve open problems posed by Benjamini and Sznitman in
arXiv:math/0610802, who studied the small u regime in high dimension. The
proofs are based on a coupling of the random walk with random interlacements on
Z^d. Among other techniques, the construction of this coupling employs a
refined use of discrete potential theory. By itself, this coupling strengthens
a result in arXiv:0802.3654.Comment: 37 pages, 3 figure
Vector-valued optimal Lipschitz extensions

Consider a bounded open set U C R[superscript n] and a Lipschitz function g : aU → R[superscript m]. Does this function always have a canonical optimal Lipschitz extension to all of U? We propose a notion of optimal Lipschitz extension and address existence and uniqueness in some special cases. In the case n = m = 2, we show that smooth solutions have two phases: in one they are conformal and in the other they are variants of infinity harmonic functions called infinity harmonic fans. We also prove existence and uniqueness for the extension problem on finite graphs.Original manuscript June 9, 201
Large Deviations for Brownian Intersection Measures

We consider $p$ independent Brownian motions in $\R^d$. We assume that $p\geq
2$ and $p(d-2)<d$. Let $\ell_t$ denote the intersection measure of the $p$
paths by time $t$, i.e., the random measure on $\R^d$ that assigns to any
measurable set $A\subset \R^d$ the amount of intersection local time of the
motions spent in $A$ by time $t$. Earlier results of Chen \cite{Ch09} derived
the logarithmic asymptotics of the upper tails of the total mass $\ell_t(\R^d)$
as $t\to\infty$. In this paper, we derive a large-deviation principle for the
normalised intersection measure $t^{-p}\ell_t$ on the set of positive measures
on some open bounded set $B\subset\R^d$ as $t\to\infty$ before exiting $B$. The
rate function is explicit and gives some rigorous meaning, in this asymptotic
regime, to the understanding that the intersection measure is the pointwise
product of the densities of the normalised occupation times measures of the $p$
motions. Our proof makes the classical Donsker-Varadhan principle for the
latter applicable to the intersection measure.
  A second version of our principle is proved for the motions observed until
the individual exit times from $B$, conditional on a large total mass in some
compact set $U\subset B$. This extends earlier studies on the intersection
measure by K\"onig and M\"orters \cite{KM01,KM05}.Comment: To appear in "Communications on Pure and Applied Mathematics
A family of non-parametric density estimation algorithms

A new methodology for density estimation is proposed. The method- ology, which builds on the one developed in [15], normalizes the data points through the composition of simple maps. The parameters of each map are determined through the maximization of a local quadratic approximation to the log-likelihood. Various candidates for the el- ementary maps of each step are proposed; criteria for choosing one includes robustness, computational simplicity and good behavior in high-dimensional settings. A good choice is that of localized radial expansions, which depend on a single parameter: all the complex- ity of arbitrary, possibly convoluted probability densities can be built through the composition of such simple maps.Fil: Tabak, E. G. . University Of New York; Estados UnidosFil: Turner, Cristina Vilma. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Córdoba. Centro de Investigación y Estudios de Matemática de Córdoba(p); Argentin
Equivariant characteristic classes of singular complex algebraic
  varieties

Homology Hirzebruch characteristic classes for singular varieties have been
recently defined by Brasselet-Schuermann-Yokura as an attempt to unify
previously known characteristic class theories for singular spaces (e.g.,
MacPherson-Chern classes, Baum-Fulton-MacPherson Todd classes, and
Goresky-MacPherson L-classes, respectively). In this note we define equivariant
analogues of these classes for singular quasi-projective varieties acted upon
by a finite group of algebraic automorphisms, and show how these can be used to
calculate the homology Hirzebruch classes of global quotient varieties. We also
compute the new classes in the context of monodromy problems, e.g., for
varieties that fiber equivariantly (in the complex topology) over a connected
algebraic manifold. As another application, we discuss Atiyah-Meyer type
formulae for twisted Hirzebruch classes of global orbifolds.Comment: v2: updates include a motivic approach, as well as an Atiyah-Meyer
  formula for global orbifolds, including a defect formul
Scaling limits of additive functionals of interacting particle systems

Em publicaçãoUsing the renormalization method introduced in [arXiv:1003.4478v1], we prove
what we call the local Boltzmann-Gibbs principle for conservative,
stationary interacting particle systems in dimension d=1. As applications of
this result, we obtain various scaling limits of additive functionals of
particle systems, like the occupation time of a given site or extensive additive
fields of the dynamics. As a by-product of these results, we also construct a
novel process, related to the stationary solution of the stochastic Burgers
equation.FC
Approximating rough stochastic PDEs

We study approximations to a class of vector-valued equations of Burgers type
driven by a multiplicative space-time white noise. A solution theory for this
class of equations has been developed recently in [Hairer, Weber, Probab.
Theory Related Fields, to appear]. The key idea was to use the theory of
controlled rough paths to give definitions of weak / mild solutions and to set
up a Picard iteration argument.
  In this article the limiting behaviour of a rather large class of (spatial)
approximations to these equations is studied. These approximations are shown to
converge and convergence rates are given, but the limit may depend on the
particular choice of approximation. This effect is a spatial analogue to the
It\^o-Stratonovich correction in the theory of stochastic ordinary differential
equations, where it is well known that different approximation schemes may
converge to different solutions.Comment: 80 pages; Corrects a mistake in the proof of Lemma 3.
Spectrum of Markov generators on sparse random graphs

Correction in Proposition 4.3. Final version.International audienceWe investigate the spectrum of the infinitesimal generator of the continuous time random walk on a randomly weighted oriented graph. This is the non-Hermitian random nxn matrix L defined by L(j,k)=X(j,k) if k<>j and L(j,j)=-sum(L(j,k),k<>j), where X(j,k) are i.i.d. random weights. Under mild assumptions on the law of the weights, we establish convergence as n tends to infinity of the empirical spectral distribution of L after centering and rescaling. In particular, our assumptions include sparse random graphs such as the oriented Erdős-Rényi graph where each edge is present independently with probability p(n)->0 as long as np(n) >> (log(n))^6. The limiting distribution is characterized as an additive Gaussian deformation of the standard circular law. In free probability terms, this coincides with the Brown measure of the free sum of the circular element and a normal operator with Gaussian spectral measure. The density of the limiting distribution is analyzed using a subordination formula. Furthermore, we study the convergence of the invariant measure of L to the uniform distribution and establish estimates on the extremal eigenvalues of L
Bogoliubov spectrum of interacting Bose gases

Final version to appear in Comm. Pure Appl. Math.International audienceWe study the large-N limit of a system of N bosons interacting with a potential of intensity 1/N. When the ground state energy is to the first order given by Hartree's theory, we study the next order, predicted by Bogoliubov's theory. We show the convergence of the lower eigenvalues and eigenfunctions towards that of the Bogoliubov Hamiltonian (up to a convenient unitary transform). We also prove the convergence of the free energy when the system is sufficiently trapped. Our results are valid in an abstract setting, our main assumptions being that the Hartree ground state is unique and non-degenerate, and that there is complete Bose-Einstein condensation on this state. Using our method we then treat two applications: atoms with ''bosonic'' electrons on one hand, and trapped 2D and 3D Coulomb gases on the other hand
Free Energy Fluctuations for Directed Polymers in Random Media in 1 + 1 Dimension

We consider two models for directed polymers in space-time independent random media (the O'Connell-Yor semidiscrete directed polymer and the continuum directed random polymer) at positive temperature and prove their KPZ universality via asymptotic analysis of exact Fredholm determinant formulas for the Laplace transform of their partition functions. In particular, we show that for large time τ, the probability distributions for the free energy fluctuations, when rescaled by τ [superscript 1 over 3], converges to the GUE Tracy-Widom distribution.
We also consider the effect of boundary perturbations to the quenched random media on the limiting free energy statistics. For the semidiscrete directed polymer, when the drifts of a finite number of the Brownian motions forming the quenched random media are critically tuned, the statistics are instead governed by the limiting Baik–Ben Arous–Péché distributions from spiked random matrix theory. For the continuum polymer, the boundary perturbations correspond to choosing the initial data for the stochastic heat equation from a particular class, and likewise for its logarithm—the Kardar-Parisi-Zhang equation. The Laplace transform formula we prove can be inverted to give the one-point probability distribution of the solution to these stochastic PDEs for the class of initial data.National Science Foundation (U.S.) (Grant DMS-1056390)National Science Foundation (U.S.) (Grant DMS-1208998)Clay Mathematics Institute (Clay Research Fellowship)Microsoft Research (Schramm Memorial Fellowship
Heat Production of Non-Interacting Fermions Subjected to Electric Fields

Electric resistance in conducting media is related to heat (or entropy)
production in presence of electric fields. In this paper, by using Araki's
relative entropy for states, we mathematically define and analyze the heat
production of free fermions within external potentials. More precisely, we
investigate the heat production of the non-autonomous C*-dynamical system
obtained from the fermionic second quantization of a discrete Schr\"odinger
operator with bounded static potential in presence of an electric field that is
time- and space-dependent. It is a first preliminary step towards a
mathematical description of transport properties of fermions from thermal
considerations. This program will be carried out in several papers. The regime
of small and slowly varying in space electric fields is important in this
context, and is studied the present paper. We use tree-decay bounds of the
$n$-point, $n\in 2\mathbb{N}$, correlations of the many-fermion system to
analyze this regime. We verify below the 1st law of thermodynamics for the
system under consideration. The latter implies, for systems doing no work, that
the heat produced by the electromagnetic field is exactly the increase of the
internal energy resulting from the modification of the (infinite volume) state
of the fermion system. The identification of heat production with an energy
increment is, among other things, technically convenient. We initially focus
our study on non-interacting (or free) fermions, but our approach will be later
applied to weakly interacting fermions
Global III-posedness of the isentropic system of gas dynamics

We consider the isentropic compressible Euler system in 2 space dimensions with pressure law p ($\rho$) = $\rho$$^2$ and we show the existence of classical Riemann data, i.e. pure jump discontinuities across a line, for which there are infinitely many admissible bounded weak solutions (bounded away from the void). We also show that some of these Riemann data are generated by a 1-dimensional compression wave: our theorem leads therefore to Lipschitz initial data for which there are infinitely many global bounded admissible weak solutions
Quantitative erivation of the Gross-Pitaevskii equation

Starting from first-principle many-body quantum dynamics, we show that the dynamics of Bose-Einstein condensates can be approximated by the time-dependent nonlinear Gross-Pitaevskii equation, giving a bound on the rate of the convergence. Initial data are constructed on the bosonic Fock space applying an appropriate Bogoliubov transformation on a coherent state with expected number of particles $\mathit{N}$. The Bogoliubov transformation plays a crucial role; it produces the correct microscopic correlations among the particles. Our analysis shows that, on the level of the one-particle reduced density, the form of the initial data is preserved by the many-body evolution, up to a small error that vanishes as $\mathit{N}^{-1/2}$ in the limit of large $\mathit{N}$
Coercivity of combined boundary integral equations in high-frequency scattering

We prove that the standard second-kind integral equation formulation of the exterior Dirichlet problem for the Helmholtz equation is coercive (i.e., sign-definite) for all smooth convex domains when the wavenumber k is sufficiently large. (This integral equation involves the so-called combined potential, or combined field, operator.) This coercivity result yields k-explicit error estimates when the integral equation is solved using the Galerkin method, regardless of the particular approximation space used (and thus these error estimates apply to several hybrid numerical-asymptotic methods developed recently). Coercivity also gives k-explicit bounds on the number of GMRES iterations needed to achieve a prescribed accuracy when the integral equation is solved using the Galerkin method with standard piecewise-polynomial subspaces. The coercivity result is obtained by using identities for the Helmholtz equation originally introduced by Morawetz in her work on the local energy decay of solutions to the wave equatio
Non equilibrium density profiles in Lorentz tubes with thermostated
  boundaries

We consider a long Lorentz tube with absorbing boundaries. Particles are
injected to the tube from the left end. We compute the equilibrium density
profiles in two cases: the semi-infinite tube (in which case the density is
constant) and a long finite tube (in which case the density is linear). In the
latter case, we also show that convergence to equilibrium is well described by
the heat equation. In order to prove these results, we obtain new results for
the Lorentz particle which are of independent interest. First, we show that a
particle conditioned not to hit the boundary for a long time converges to the
Brownian meander. Second, we prove several local limit theorems for particles
having a prescribed behavior in the past
A quantum dynamical approach to matrix Khrushchev's formulas

Khrushchev's formula is the cornerstone of the so called Khrushchev theory, a
body of results which has revolutionized the theory of orthogonal polynomials
on the unit circle. This formula can be understood as a factorization of the
Schur function for an orthogonal polynomial modification of a measure on the
unit circle. No such formula is known in the case of matrix-valued measures.
This constitutes the main obstacle to generalize Khrushchev theory to the
matrix-valued setting which we overcome in this paper.
  It was recently discovered that orthogonal polynomials on the unit circle and
their matrix-valued versions play a significant role in the study of quantum
walks, the quantum mechanical analogue of random walks. In particular, Schur
functions turn out to be the mathematical tool which best codify the return
properties of a discrete time quantum system, a topic in which Khrushchev's
formula has profound and surprising implications. We will show that this
connection between Schur functions and quantum walks is behind a simple proof
of Khrushchev's formula via `quantum' diagrammatic techniques for CMV matrices.
This does not merely give a quantum meaning to a known mathematical result,
since the diagrammatic proof also works for matrix-valued measures. Actually,
this path counting approach is so fruitful that it provides different matrix
generalizations of Khrushchev's formula, some of them new even in the case of
scalar measures.
  Furthermore, the path counting approach allows us to identify the properties
of CMV matrices which are responsible for Khrushchev's formula. On the one
hand, this helps to formalize and unify the diagrammatic proofs using simple
operator theory tools. On the other hand, this is the origin of our main result
which extends Khrushchev's formula beyond the CMV case, as a factorization rule
for Schur functions related to general unitary operators.Comment: 41 page
Mean-field evolution of fermionic mixed states

In this paper we study the dynamics of fermionic mixed states in the mean-field regime. We consider initial states that are close to quasi-free states and prove that, under suitable assumptions on the initial data and on the many-body interaction, the quantum evolution of such initial data is well approximated by a suitable quasi-free state. In particular, we prove that the evolution of the reduced one-particle density matrix converges, as the number of particles goes to infinity, to the solution of the time-dependent Hartree-Fock equation. Our result holds for all times and gives effective estimates on the rate of convergence of the many-body dynamics towards the Hartree-Fock evolution
Shear Flows of an Ideal Fluid and Elliptic Equations in Unbounded Domains

International audienceWe prove that, in a two-dimensional strip, a steady flow of an ideal incompressible fluid with no stationary point and tangential boundary conditions is a shear flow. The same conclusion holds for a bounded steady flow in a half-plane. The proofs are based on the study of the geometric properties of the streamlines of the flow and on one-dimensional symmetry results for solutions of some semilinear elliptic equations. Some related rigidity results of independent interest are also shown in n-dimensional slabs in any dimension n