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High-dimensional linear regression has been intensively studied in the community of statistics in the last two decades. For the convenience of theoretical analyses, classical methods usually assume independent observations and sub-Gaussian-tailed errors. However, neither of them hold in many real high-dimensional time-series data. Recently [Sun, Zhou, Fan, 2019, J. Amer. Stat. Assoc., in press] proposed Adaptive Huber Regression (AHR) to address the issue of heavy-tailed errors. They discover that the robustification parameter of the Huber loss should adapt to the sample size, the dimensionality, and the moments of the heavy-tailed errors. We progress in a vertical direction and justify AHR on dependent observations. Specifically, we consider an important dependence structure -- Markov dependence. Our results show that the Markov dependence impacts on the adaption of the robustification parameter and the estimation of regression coefficients in the way that the sample size should be discounted by a factor depending on the spectral gap of the underlying Markov chain.

We construct a generalisation of Bakry-Emery curvature to prove twisted intertwining relations for Markov semigroups. These relations are applied to Brascamp–Lieb type inequalities and spectral gap results. It extends the method of Arnaudon, Bonnefont and Joulin, to Riemannian manifolds and to a wider class of twists. These results are illustrated with several examples.

The paper gives sharp spectral gap conditions for existence of inertial manifolds for abstract semilinear parabolic equations with non-self-adjoint leading part. Main attention is paid to the case where this leading part have Jordan cells which appear after applying the so-called Kwak transform to various important equations such as 2D Navier–Stokes equations, reaction-diffusion-advection systems, etc. The different forms of Kwak transforms and relations between them are also discussed.

We give a two sided estimate on the spectral gap for the Boltzmann measures μh on the circle. We prove that the spectral gap is greater than 1 for any h ∈ ℝ and the spectral gap tends to the positive infinity as h → ∞ with speed ∣h∣.

We consider the stationary O’Connell–Yor model of semi-discrete directed polymers in a Brownian environment in the intermediate disorder regime and show convergence of the increments of the log-partition function to the energy solutions of the stochastic Burgers equation. The proof does not rely on the Cole–Hopf transform and avoids the use of spectral gap estimates for the discrete model. The key technical argument is a second-order Boltzmann–Gibbs principle.

Ergodicity of random dynamical systems with a periodic measure is obtained on a Polish space. In the Markovian case, the idea of Poincar\'e sections is introduced. It is proved that if the periodic measure is PS-ergodic, then it is ergodic. Moreover, if the infinitesimal generator of the Markov semigroup only has equally placed simple eigenvalues including $0$ on the imaginary axis, then the periodic measure is PS-ergodic and has positive minimum period. Conversely if the periodic measure with the positive minimum period is PS-mixing, then the infinitesimal generator only has equally placed simple eigenvalues (infinitely many) including $0$ on the imaginary axis. Moreover, under the spectral gap condition, PS-mixing of the periodic measure is proved. The ``equivalence" of random periodic processes and periodic measures is established. This is a new class of ergodic random processes. Random periodic paths of stochastic perturbation of the periodic motion of an ODE is obtained., Comment: 38 pages

We are mainly concerned with sequences of graphs having a grid geometry, with a uniform local structure in a bounded domain $${\Omega} {\subset} \mathbb{R}^{d}, d \geq 1$$ Ω ⊂ R d , d ≥ 1 . When $$\Omega = [0, 1]$$ Ω = [ 0 , 1 ] , such graphs include the standard Toeplitz graphs and, for $$\Omega = [0, 1]^{d}$$ Ω = [ 0 , 1 ] d , the considered class includes d-level Toeplitz graphs. In the general case, the underlying sequence of adjacency matrices has a canonical eigenvalue distribution, in the Weyl sense, and we show that we can associate to it a symbol $$\mathfrak{f}$$ f . The knowledge of the symbol and of its basic analytical features provides many information on the eigenvalue structure, of localization, spectral gap, clustering, and distribution type.Few generalizations are also considered in connection with the notion of generalized locally Toeplitz sequences and applications are discussed, stemming e.g. from the approximation of differential operators via numerical schemes. Nevertheless, more applications can be taken into account, since the results presented here can be applied as well to study the spectral properties of adjacency matrices and Laplacian operators of general large graphs and networks.

In their previous work, the authors studied the abelian sandpile model on graphs constructed from a growing piece of a plane or space tiling, given periodic or open boundary conditions, and identified spectral factors which govern the asymptotic spectral gap and asymptotic mixing time. This article gives a general method of determining the spectral factor either computationally or asymptotically and performs the determination in specific examples., This is the second part of the original manuscript arXiv:1902.04174, which contains numerical calculation of the spectral factors

The Boltzmann equation without the angular cutoff is considered when the initial data is a small perturbation of a global Maxwellian and decays algebraically in the velocity variable. We obtain a well-posedness theory in the perturbative framework for both mild and strong angular singularities. The three main ingredients in the proof are the moment propagation, the spectral gap of the linearized operator, and the regularizing effect of the linearized operator when the initial data is in a Sobolev space with a negative index. A carefully designed pseudo-differential operator plays a central role in capturing the regularizing effect. In addition, some intrinsic symmetry with respect to the collision operator and an intrinsic functional in the coercivity estimate are essentially used in the commutator estimates for the collision operator with velocity weights.

Let G = SL(2, ℤ) ⋉ ℤ2 and H = SL(2, ℤ). We prove that the action G ↷ ℝ2 is uniformly non-amenable and that the quasi-regular representation of G on l2(G/H) has a uniform spectral gap. Both results are a consequence of a uniform quantitative form of ping-pong for affine transformations, which we establish here.

We study a $1$-dimensional chain of $N$ weakly anharmonic classical oscillators coupled at its ends to heat baths at different temperatures. Each oscillator is subject to pinning potential and it also interacts with its nearest neighbors. In our set up both potentials are homogeneous and bounded (with $N$ dependent bounds) perturbations of the harmonic ones. We show how a generalized version of Bakry-Emery theory can be adapted to this case of a hypoelliptic generator which is inspired by F. Baudoin (2017). By that we prove exponential convergence to non-equilibrium steady state (NESS) in Wasserstein-Kantorovich distance and in relative entropy with quantitative rates. We estimate the constants in the rate by solving a Lyapunov-type matrix equation and we obtain that the exponential rate, for the homogeneous chain, has order bigger than $N^{-3}$. For the purely harmonic chain the order of the rate is in $ [N^{-3},N^{-1}]$. This shows that, in this set up, the spectral gap decays at most polynomially with $N$., Comment: 54 pages, 1 figure, comments are welcome

For one-dimensional non-selfadjoint Schrodinger and Dirac operators with periodic complex-valued potentials belonging to the class $$L_2 $$, asymptotic representations for spectral gaps are obtained in terms of Fourier coefficients of the potentials and estimates for the gap lengths are given.

Von Neumann's original proof of the ergodic theorem is revisited. A uniform convergence rate is established under the assumption that one can control the density of the spectrum of the underlying self-adjoint operator when restricted to suitable subspaces. Explicit rates are obtained when the bound is polynomial, with applications to the linear Schr\"odinger and wave equations. In particular, decay estimates for time-averages of solutions are shown., Comment: 13 pages

Consider the semilinear Schrodinger equation { − △ u + V ( x ) u = f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , where both V ( x ) and f ( x , u ) are periodic in x, 0 belongs to a spectral gap of − △ + V , and f ( x , u ) is subcritical and allowed to be super-linear at some x ∈ R N and asymptotically linear at the other x ∈ R N . In the existing works in the literature, it is commonly assumed that lim | u | → ∞ ⁡ ∫ 0 u f ( x , s ) d s u 2 = ∞ uniformly in x ∈ R N , to obtain the existence of ground state solutions or infinitely many geometrically distinct solutions. In this paper, for the first time, we prove the existence of ground state solutions and infinitely many geometrically distinct solutions under the weaker super-quadratic condition lim | u | → ∞ ⁡ ∫ 0 u f ( x , s ) d s u 2 = ∞ , a.e. x ∈ G just for some domain G ⊂ R N .

We study limit theorems for partial sums of instantaneous functions of a homogeneous Markov chain on a general state space. The summands are heavy-tailed and the limits are stable distributions. The conditions imposed on the transition operator $P$ of the Markov chain ensure that the limit is the same as if the summands were independent. Such a~scheme admits a physical interpretation, as given in Jara et al. (Ann. Appl. Probab., 19 (2009), 2270--2300). We considerably extend the results of Jara et al., (ibid.) and Cattiaux and Manou-Abi (ESAIM Probab. Stat., 18 (2014), 468--486). We show that the theory holds under the assumption of operator uniform integrability in $L^2$ of $P$ (a notion introduced by Wu (J. Funct. Anal., 172 (2000), 301--376)) plus the $L^2$-spectral gap property. If we strengthen the uniform integrability in $L^2$ to the hyperboundedness, then the $L^2$-spectral gap property can be relaxed to the strong mixing at geometric rate (in practice: to geometric ergodicity). We provide an example of a Markov chain on a countable space that is uniformly integrable in $L^2$ (and admits an $L^2$-spectral gap), while it is not hyperbounded. Moreover, we show by example that hyperboundedness is still a weaker property than $\phi$-mixing, what enlarges the range of models of interest. What makes our assumptions working is a new, efficient version of the Principle of Conditioning that operates with conditional characteristic functions rather than predictable characteristics., Comment: 29 pages

We study quantum chains whose Hamiltonians are perturbations by bounded interactions of short range of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. We prove that, for small values of a coupling constant, the spectral gap of the perturbed Hamiltonian above its ground-state energy is bounded from below by a positive constant uniformly in the length of the chain. In our proof we use a novel method based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain.

Tensor network states are used extensively as a mathematically convenient description of physically relevant states of many-body quantum systems. Those built on regular lattices, i.e. matrix product states (MPS) in dimension 1 and projected entangled pair states (PEPS) in dimension 2 or higher, are of particular interest in condensed matter physics. The general goal of this work is to characterize which features of MPS and PEPS are generic and which are, on the contrary, exceptional. This problem can be rephrased as follows: given an MPS or PEPS sampled at random, what are the features that it displays with either high or low probability? One property which we are particularly interested in is that of having either rapidly decaying or long-range correlations. In a nutshell, our main result is that translation-invariant MPS and PEPS typically exhibit exponential decay of correlations at a high rate. We have two distinct ways of getting to this conclusion, depending on the dimensional regime under consideration. Both yield intermediate results which are of independent interest, namely: the parent Hamiltonian and the transfer operator of such MPS and PEPS typically have a large spectral gap. In all these statements, our aim is to get a quantitative estimate of the considered quantity (generic correlation length or spectral gap), which has the best possible dependency on the physical and bond dimensions of the random MPS or PEPS., 53 pages, 7 figures. v2: a bug in Section 4 has been fixed and a few changes have been made to the presentation. v3: published version

International audience; Interpolation inequalities play an essential role in Analysis with fundamental consequences in Mathematical Physics, Nonlinear Partial Differential Equations (PDEs), Markov Processes, etc., and have a wide range of applications in various other areas of Science. Research interests have evolved over the years: while mathematicians were originally focussed on abstract properties (for instance appropriate notions of functional spaces for the existence of weak solutions in PDEs), more qualitative questions (for instance, bifurcation diagrams, multiplicity of the solutions in PDEs and their qualitative behaviour) progressively emerged. The use of entropy methods in nonlinear PDEs is a typical example: in some cases, the optimal constant in the inequality can be interpreted as an optimal rate of decay of an entropy for an associated evolution equation. Much more has been learned by adopting this point of view. This paper aims at illustrating some of these recent aspect of entropyentropy production inequalities, with applications to stability in Gagliardo-Nirenberg-Sobolev inequalities and symmetry results in Caffarelli-Kohn-Nirenberg inequalities. Entropy methods provide a framework which relates nonlinear regimes with their linearized counterparts. This framework allows to prove optimality results, symmetry results and stability estimates. Some emphasis will be put on the hidden structure which explain such properties. Related open problems will be listed.

In this work, we establish L2-exponential convergence for a broad class of Piecewise Deterministic Markov Processes recently proposed in the context of Markov Process Monte Carlo methods and covering in particular the Randomized Hamiltonian Monte Carlo, the Zig-Zag process and the Bouncy Particle Sampler. The kernel of the symmetric part of the generator of such processes is non-trivial, and we follow the ideas recently introduced by (Dolbeault et al., 2009, 2015) to develop a rigorous framework for hypocoercivity in a fairly general and unifying set-up, while deriving tractable estimates of the constants involved in terms of the parameters of the dynamics. As a by-product we characterize the scaling properties of these algorithms with respect to the dimension of classes of problems, therefore providing some theoretical evidence to support their practical relevance.

In recent work by Einsiedler, Mozes, Shah and Shapira the limiting distributions of primitive rational points on expanding horospheres was examined in arbitrary dimension, and a suspended version of this result was announced. Motivated by this, we prove an analog to the announcement for primitive rational points in Hilbert modular surfaces via effective mixing., Comment: Corrections to Propositions 3.5 and 3.6. General improvements on exposition and notation

We study dynamical optimal transport metrics between density matrices associated to symmetric Dirichlet forms on finite-dimensional $C^*$-algebras. Our setting covers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein--Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, and spectral gap estimates., An error in Theorem 10.6 has been corrected

The asymptotics of eigenvalues appearing near the lower edge of a spectral gap of the Dirichlet problem is studied for the Laplace operator in a d-dimensional periodic waveguide with a singular perturbation of the boundary by creating a hole with a small diameter e. Several versions of the structure of the gap edge are considered. As usual, the asymptotic formulas are different in the cases d ≥ 3 and d = 2, where the eigenvalues occur at distances O(e2(d−2)) or O(e2d) and O(|ln e|−2) or O(e4), respectively, from the gap edge. Other types of singular perturbation of the waveguide surface and other types of boundary conditions are discussed, which provide the appearance of eigenvalues near both edges of one or several gaps.

The abelian sandpile model defines a Markov chain whose states are integer-valued functions on the vertices of a simple connected graph G G . By viewing this chain as a (nonreversible) random walk on an abelian group, we give a formula for its eigenvalues and eigenvectors in terms of “multiplicative harmonic functions” on the vertices of G G . We show that the spectral gap of the sandpile chain is within a constant factor of the length of the shortest noninteger vector in the dual Laplacian lattice, while the mixing time is at most a constant times the smoothing parameter of the Laplacian lattice. We find a surprising inverse relationship between the spectral gap of the sandpile chain and that of simple random walk on G G : If the latter has a sufficiently large spectral gap, then the former has a small gap! In the case where G G is the complete graph on n n vertices, we show that the sandpile chain exhibits cutoff at time 1 4 π 2 n 3 log ⁡ n \frac {1}{4\pi ^{2}}n^{3}\log n .

Let $\Gamma _2\subseteq \Gamma _1$ be finitely generated subgroups of ${\operatorname{GL}}_{n_0}({\mathbb{Z}}[1/q_0])$ where $q_0$ is a positive integer. For $i=1$ or $2$, let ${\mathbb{G}}_i$ be the Zariski-closure of $\Gamma _i$ in $({\operatorname{GL}}_{n_0})_{{\mathbb{Q}}}$, ${\mathbb{G}}_i^{\circ }$ be the Zariski-connected component of ${\mathbb{G}}_i$, and let $G_i$ be the closure of $\Gamma _i$ in $\prod _{p\nmid q_0}{\operatorname{GL}}_{n_0}({\mathbb{Z}}_p)$. In this article we prove that if ${\mathbb{G}}_1^{\circ }$ is the smallest closed normal subgroup of ${\mathbb{G}}_1^{\circ }$ that contains ${\mathbb{G}}_2^{\circ }$ and $\Gamma _2\curvearrowright G_2$ has spectral gap, then $\Gamma _1\curvearrowright G_1$ has spectral gap.

The purpose of this survey is to discuss Poisson and contact brackets and related infinite dimensional superalgebras. All vector spaces are considered over the field of complex numbers $${\mathbb {C}}$$ .

We give a non-trivial upper bound for the critical density when stabilizing i.i.d. distributed sandpiles on the lattice $\mathbb{Z}^2$. We also determine the asymptotic spectral gap, asymptotic mixing time and prove a cutoff phenomenon for the recurrent state abelian sandpile model on the torus $\left( \mathbb{Z} / m\mathbb{Z} \right)^2$. The techniques use analysis of the space of functions on $\mathbb{Z}^2$ which are harmonic modulo 1. In the course of our arguments, we characterize the harmonic modulo 1 functions in $\ell^p(\mathbb{Z}^2)$ as linear combinations of certain discrete derivatives of Green's functions, extending a result of Schmidt and Verbitskiy., Comment: v3: Minor changes

Kazhdan's property (T) has been studied for several discrete group-like structures, including standard invariants of Jones' subfactors and discrete quantum groups. We prove a Zuk-type spectral gap criterion for property (T) in this setting. This allows us to construct a family of property (T) discrete quantum groups, as well as subfactor standard invariants, given by generators and a triangle presentation. These are the first examples of property (T) discrete quantum groups that are not directly related to a discrete group with property (T)., Comment: v2: Minor changes, final version, to appear in Advances in Mathematics

We consider the trajectory of a tracer that is the solution of an ordinary differential equation $\dot\bbX(t)=\bbV(t, \bbX(t)),\ X(0)=0$, with the right hand side, that is a stationary, zero-mean, Gaussian vector field with incompressible realizations. It is known, see [K-F;C-X;K-L-O], that $\bbX(t)/\sqrt{t}$ converges in law, as $t\to+\infty$, to a normal, zero mean vector, provided that the field $V(t,x)$ is Markovian and has the spectral gap property. We wish to extend this result to the case when the field is not Markovian and its covariance matrix is given by a completely monotone Bernstein function., 10 pages

Let $G$ be a locally compact amenable group. We say that G has property (M) if every closed subgroup of finite covolume in G is cocompact. A classical theorem of Mostow ensures that connected solvable Lie groups have property (M). We prove a non-Archimedean extension of Mostow's theorem by showing the amenable linear locally compact groups have property (M). However property (M) does not hold for all solvable locally compact groups: indeed, we exhibit an example of a metabelian locally compact group with a non-uniform lattice. We show that compactly generated metabelian groups, and more generally nilpotent-by-nilpotent groups, do have property (M). Finally, we highlight a connection of property (M) with the subtle relation between the analytic notions of strong ergodicity and the spectral gap., Comment: 38 pages

The paper gives a comprehensive study of Inertial Manifolds for hyperbolic relaxations of an abstract semilinear parabolic equation in a Hilbert space. A new scheme of constructing Inertial Manifolds for such type of problems is suggested and optimal spectral gap conditions which guarantee their existence are established. Moreover, the dependence of the constructed manifolds on the relaxation parameter in the case of the parabolic singular limit is also studied. Bibliography: 38 titles.

We study the simultaneous exact controllability of two vibrating strings with variable physical coefficients and controlled from a common endpoint. We give sufficient conditions on the physical coefficients for which the eigenfrequencies of both systems do not coincide and the associated spectral gap is uniformly positive. Under these conditions, we show that these systems are simultaneously exactly controllable.

Several techniques of generating random quantum channels, which act on the set of $d$-dimensional quantum states, are investigated. We present three approaches to the problem of sampling of quantum channels and show under which conditions they become mathematically equivalent, and lead to the uniform, Lebesgue measure on the convex set of quantum operations. We compare their advantages and computational complexity and demonstrate which of them is particularly suitable for numerical investigations. Additional results focus on the spectral gap and other spectral properties of random quantum channels and their invariant states. We compute mean values of several quantities characterizing a given quantum channel, including its unitarity, the average output purity and the $2$-norm coherence of a channel, averaged over the entire set of the quantum channels with respect to the uniform measure. An ensemble of classical stochastic matrices obtained due to super-decoherence of random quantum stochastic maps is analyzed and their spectral properties are studied using the Bloch representation of a classical probability vector., 29 pages, 7 figures

The second largest eigenvalue of a transition matrix $P$ has connections with many properties of the underlying Markov chain, and especially its convergence rate towards the stationary distribution. In this paper, we give an asymptotic upper bound for the second eigenvalue when $P$ is the transition matrix of the simple random walk over a random directed graph with given degree sequence. This is the first result concerning the asymptotic behavior of the spectral gap for sparse non-reversible Markov chains with an unknown stationary distribution. An immediate consequence of our result is a proof of the Alon conjecture for directed regular graphs. Our result is based on a variation of the trace method introduced by Bordenave (2015)., 41 pages, with 10 figures. Second version : slight simplifications of the proof, bibliography extended. Comments are welcome

We prove a sharp general inequality estimating the distance of two probability measures on a compact Lie group in the Wasserstein metric in terms of their Fourier transforms. We use a generalized form of the Wasserstein metric, related by Kantorovich duality to the family of functions with an arbitrarily prescribed modulus of continuity. The proof is based on smoothing with a suitable kernel, and a Fourier decay estimate for continuous functions. As a corollary, we show that the rate of convergence of random walks on semisimple groups in the Wasserstein metric is necessarily almost exponential, even without assuming a spectral gap. Applications to equidistribution and empirical measures are also given.

In the semiclassical limit$$\hbar \rightarrow 0$$ħ→0, we analyze a class of self-adjoint Schrödinger operators$$H_\hbar = \hbar ^2 L + \hbar W + V\cdot {\mathrm {id}}_{\mathscr {E}}$$Hħ=ħ2L+ħW+V·idEacting on sections of a vector bundle$${\mathscr {E}}$$Eover an oriented Riemannian manifoldMwhereLis a Laplace type operator,Wis an endomorphism field and the potential energyVhas non-degenerate minima at a finite number of points$$m^1,\ldots m^r \in M$$m1,…mr∈M, called potential wells. Using quasimodes of WKB-type near$$m^j$$mjfor eigenfunctions associated with the low lying eigenvalues of$$H_\hbar $$Hħ, we analyze the tunneling effect, i.e. the splitting between low lying eigenvalues, which e.g. arises in certain symmetric configurations. Technically, we treat the coupling between different potential wells by an interaction matrix and we consider the case of a single minimal geodesic (with respect to the associated Agmon metric) connecting two potential wells and the case of a submanifold of minimal geodesics of dimension$$\ell + 1$$ℓ+1. This dimension$$\ell $$ℓdetermines the polynomial prefactor for exponentially small eigenvalue splitting.

The problem of data-driven identification of coherent observables of measure-preserving, ergodic dynamical systems is studied using kernel integral operator techniques. An approach is proposed whereby complex-valued observables with approximately cyclical behavior are constructed from a pair eigenfunctions of integral operators built from delay-coordinate mapped data. It is shown that these observables are $\epsilon$-approximate eigenfunctions of the Koopman evolution operator of the system, with a bound $\epsilon$ controlled by the length of the delay-embedding window, the evolution time, and appropriate spectral gap parameters. In particular, $ \epsilon$ can be made arbitrarily small as the embedding window increases so long as the corresponding eigenvalues remain sufficiently isolated in the spectrum of the integral operator. It is also shown that the time-autocorrelation functions of such observables are $\epsilon$-approximate Koopman eigenvalue, exhibiting a well-defined characteristic oscillatory frequency (estimated using the Koopman generator) and a slowly-decaying modulating envelope. The results hold for measure-preserving, ergodic dynamical systems of arbitrary spectral character, including mixing systems with continuous spectrum and no non-constant Koopman eigenfunctions in $L^2$. Numerical examples reveal a coherent observable of the Lorenz 63 system whose autocorrelation function remains above 0.5 in modulus over approximately 10 Lyapunov timescales., Comment: 37 pages, 6 figures

The celebrated Littlewood-Offord problem asks for an upper bound on the probability that the random variable $\epsilon_1 v_1 + \cdots + \epsilon_n v_n$ lies in the Euclidean unit ball, where $\epsilon_1, \ldots, \epsilon_n \in \{-1, 1\}$ are independent Rademacher random variables and $v_1, \ldots, v_n \in \mathbb{R}^d$ are fixed vectors of at least unit length.We extend many known results to the case that the $\epsilon_i$ are obtained from a Markov chain, including the general bounds first shown by Erd\H{o}s in the scalar case and Kleitman in the vector case, and also under the restriction that the $v_i$ are distinct integers due to S\'ark\"ozy and Szemeredi. In all extensions, the upper bound includes an extra factor depending on the spectral gap. We also construct a pseudorandom generator for the Littlewood-Offord problem using similar techniques.

We study the stability with respect to a broad class of perturbations of gapped ground state phases of quantum spin systems defined by frustration-free Hamiltonians. The core result of this work is a proof using the Bravyi-Hastings-Michalakis (BHM) strategy that under a condition of Local Topological Quantum Order, the bulk gap is stable under perturbations that decay at long distances faster than a stretched exponential. Compared to previous work we expand the class of frustration-free quantum spin models that can be handled to include models with more general boundary conditions, and models with discrete symmetry breaking. Detailed estimates allow us to formulate sufficient conditions for the validity of positive lower bounds for the gap that are uniform in the system size and that are explicit to some degree. We provide a survey of the BHM strategy following the approach of Michalakis and Zwolak, with alterations introduced to accommodate more general than just periodic boundary conditions and more general lattices. We express the fundamental condition known as LTQO by means of the notion of indistinguishability radius, which we introduce. Using the uniform finite-volume results we then proceed to study the thermodynamic limit. We first study the case of a unique limiting ground state and then also consider models with spontaneous breaking of a discrete symmetry. In the latter case, LTQO cannot hold for all local observables. However, for perturbations that preserve the symmetry, we show stability of the gap and the structure of the broken symmetry phases. We prove that the GNS Hamiltonian associated with each pure state has a non-zero spectral gap above the ground state., Turned off `showkeys' :) and minor edits

We study an effective Hamiltonian for the standard $$\nu =1/3$$ ν = 1 / 3 fractional quantum Hall system in the thin cylinder regime. We give a complete description of its ground state space in terms of what we call Fragmented Matrix Product States, which are labeled by a certain family of tilings of the one-dimensional lattice. We then prove that the model has a spectral gap above the ground states for a range of coupling constants that includes physical values. As a consequence of the gap we establish the incompressibility of the fractional quantum Hall states. We also show that all the ground states labeled by a tiling have a finite correlation length, for which we give an upper bound. We demonstrate by example, however, that not all superpositions of tiling states have exponential decay of correlations.

Considerons une matrice bi-stochastique aleatoire de taille $n$ et de la forme $MQ$ avec $M$ une matrice de permutation uniformement distribuee et $Q$ une matrice bi-stochastique fixee. Sous des conditions de parcimonie et de regularite sur $Q$, on demontre que la deuxieme plus grande valeur propre de $MQ$ est essentiellement bornee par la norme de Hilbert–Schmidt normalisee de $Q$ lorsque $n$ est tres grand. Ce resultat s’applique aux marches au hasard sur les graphes aleatoires diriges reguliers.

We develop a method for producing estimates on the spectral gaps of reversible Markov jump processes with chaotic invariant measures, that is effective in the case of degenerate jump rates, and we apply it to prove the Kac conjecture for hard sphere collision in three dimensions.

Kinetically constrained models (KCM) are reversible interacting particle systems on $\mathbb{Z}^d$ with continuous-time constrained Glauber dynamics. They are a natural non-monotone stochastic version of the family of cellular automata with random initial state known as $\mathcal{U}$-bootstrap percolation. KCM have an interest in their own right, owing to their use for modelling the liquid-glass transition in condensed matter physics. In two dimensions there are three classes of models with qualitatively different scaling of the infection time of the origin as the density of infected sites vanishes. Here we study in full generality the class termed `critical'. Together with the companion paper by Martinelli and two of the authors we establish the universality classes of critical KCM and determine within each class the critical exponent of the infection time as well as of the spectral gap. In this work we prove that for critical models with an infinite number of stable directions this exponent is twice the one of their bootstrap percolation counterpart. This is due to the occurrence of `energy barriers', which determine the dominant behaviour for these KCM but which do not matter for the monotone bootstrap dynamics. Our result confirms the conjecture of Martinelli, Morris and the last author, who proved a matching upper bound., 33 pages, 6 figures, expanded section 5.1, minor improvements of the presentation

International audience; By viewing the $N$-simplex as the set of positions of $N-1$ ordered particles on the unit interval, the adjacent walk is the continuous time Markov chain obtained by updating independently at rate 1 the position of each particle with a sample from the uniform distribution over the interval given by the two particles adjacent to it. We determine its spectral gap and prove that both the total variation distance and the separation distance to the uniform distribution exhibit a cutoff phenomenon, with mixing times that differ by a factor $2$. The results are extended to the family of log-concave distributions obtained by replacing the uniform sampling by a symmetric log-concave Beta distribution.

In this paper, we study the second-order Hamiltonian systems $$ \ddot{u}-L(t)u+\nabla W(t,u)=0, $$ u ¨ − L ( t ) u + ∇ W ( t , u ) = 0 , where $t\in \mathbb{R}$ t ∈ R , $u\in \mathbb{R}^{N}$ u ∈ R N , L and W depend periodically on t, 0 lies in a spectral gap of the operator $-d^{2}/dt^{2}+L(t)$ − d 2 / d t 2 + L ( t ) and $W(t,x)$ W ( t , x ) is locally superquadratic. Replacing the common superquadratic condition that $\lim_{|x|\rightarrow \infty }\frac{W(t,x)}{|x|^{2}}=+\infty $ lim | x | → ∞ W ( t , x ) | x | 2 = + ∞ uniformly in $t\in \mathbb{R}$ t ∈ R by the local condition that $\lim_{|x|\rightarrow \infty }\frac{W(t,x)}{|x|^{2}}=+\infty $ lim | x | → ∞ W ( t , x ) | x | 2 = + ∞ a.e. $t\in J$ t ∈ J for some open interval $J\subset \mathbb{R}$ J ⊂ R , we prove the existence of one nontrivial homoclinic soluiton for the above problem.

In this paper, we study random walks on dynamical random environments in $1+1$ dimensions. Assuming that the environment is invariant under space-time shifts and fulfills a mild mixing hypothesis, we establish a law of large numbers and a concentration inequality around the asymptotic speed. The mixing hypothesis imposes a polynomial decay rate of covariances on the environment with sufficiently high exponent but does not impose uniform mixing. Examples of environments for which our methods apply include the contact process and Markovian environments with a positive spectral gap, such as the East model. For the East model, we also obtain that the distinguished zero satisfies a law of large numbers with strictly positive speed.

The emergence of big data has led to so-called convergence complexity analysis, which is the study of how Markov chain Monte Carlo (MCMC) algorithms behave as the sample size, $n$, and/or the number of parameters, $p$, in the underlying data set increase. This type of analysis is often quite challenging, in part because existing results for fixed $n$ and $p$ are simply not sharp enough to yield good asymptotic results. One of the first convergence complexity results for an MCMC algorithm on a continuous state space is due to Yang and Rosenthal (2019), who established a mixing time result for a Gibbs sampler (for a simple Bayesian random effects model) that was introduced and studied by Rosenthal (1996). The asymptotic behavior of the spectral gap of this Gibbs sampler is, however, still unknown. We use a recently developed simulation technique (Qin et. al., 2019) to provide substantial numerical evidence that the gap is bounded away from 0 as $n \rightarrow \infty$. We also establish a pair of rigorous convergence complexity results for two different Gibbs samplers associated with a generalization of the random effects model considered by Rosenthal (1996). Our results show that, under strong regularity conditions, the spectral gaps of these Gibbs samplers converge to 1 as the sample size increases., Revision resubmitted to Methodology and Computing in Applied Probability, 34 pages, 1 figure

The work of the first and third authors was supported by the project MTM2017- 85067-P, funded by the Spanish government and the European Regional Development Fund and they gratefully acknowledge the support of the Hausdorff Research Institute for Mathematics (Bonn), through the Junior Trimester Program on Kinetic Theory. The work of the second author was supported by the ANR project NOLO, grant ANR-20-CE40-0015, funded by the French Ministry of Research. The work of the third author was also supported by the Basque Government through the BERC 2018-2021 program, by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2017-0718, by the ``la Caixa"" Foundation, and by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme grant 639638., We study the long-time behavior of the growth-fragmentation equation, a nonlocal linear evolution equation describing a wide range of phenomena in structured population dynamics. We show the existence of a spectral gap under conditions that generalize those in the literature by using a method based on Harris's theorem, a result coming from the study of equilibration of Markov processes. The difficulty posed by the nonconservativeness of the equation is overcome by performing an h-transform, after solving the dual Perron eigenvalue problem. The existence of the direct Perron eigenvector is then a consequence of our methods, which prove exponential contraction of the evolution equation. Moreover the rate of convergence is explicitly quantifiable in terms of the dual eigenfunction and the coefficients of the equation., BERC, French Ministry of Research, Hausdorff Research Institute for Mathematics ANR-20-CE40-0015, Spanish government, European Commission 639638 EC, European Research Council, Ministerio de Economía y Competitividad ¡SEV-2017-0718 MINECO, European Regional Development Fund

International audience; Under the standard drift/minorization and strong aperiodicity assumptions, this paper provides an original and quite direct approach of the V-geometrical ergodicity of a general Markov kernel P , which is by now a classical framework in Markov modelling. This is based on an explicit approximation of the iterates of P by positive finite-rank operators, combined with the Krein-Rutman theorem in its version on topological dual spaces. Moreover this allows us to get a new bound on the spectral gap of the transition kernel. This new approach is expected to shed new light on the role and on the interest of the above mentioned drift/minorization and strong aperiodicity assumptions in V-geometrical ergodicity.

Let [Formula: see text] be a group and [Formula: see text] be a descending sequence of finite-index normal subgroups. We establish explicit upper bounds on the diameters of the directed Cayley graphs of the [Formula: see text], under some natural hypotheses on the behavior of power and commutator words in [Formula: see text]. The bounds we obtain do not depend on a choice of generating set. Moreover, under reasonable conditions our method provides a fast algorithm for navigating directed Cayley graphs. The proof is closely analogous to the Solovay–Kitaev procedure, which only uses commutator words, but also only constructs small-diameter undirected Cayley graphs. We apply our procedure to give new directed diameter bounds on finite quotients of a large class of regular branch groups, and of [Formula: see text] (for [Formula: see text] even).

We consider a natural local dynamic on the set of all rooted planar maps with $n$ edges that is in some sense analogous to “edge flip” Markov chains, which have been considered before on a variety of combinatorial structures (triangulations of the $n$-gon and quadrangulations of the sphere, among others). We provide the first polynomial upper bound for the mixing time of this “edge rotation” chain on planar maps: we show that the spectral gap of the edge rotation chain is bounded below by an appropriate constant times $n^{-11/2}$. In doing so, we provide a partially new proof of the fact that the same bound applies to the spectral gap of edge flips on quadrangulations as defined in [8], which makes it possible to generalise the result of [8] to a variant of the edge flip chain related to edge rotations via Tutte’s bijection.