Spectral Gap Research Articles

Thermalization in Kitaev’s quantum double models via tensor network techniques

Abstract We show that every ergodic Davies generator associated to any 2D Kitaev’s quantum double model has a nonvanishing spectral gap in the thermodynamic limit. This validates rigorously the extended belief that those models are useless as self-correcting quantum memories, even in the non-abelian case. The proof uses recent ideas and results regarding the characterization of the spectral gap for parent Hamiltonians associated to Projected Entangled Pair States in terms of a bulk-boundary correspondence.

Hairer-Quastel universality in non-stationarity via energy solution theory

The paper addresses probabilistic aspects of the KPZ equation and stochastic Burgers equation by providing a solution theory that builds on the energy solution theory [15, 19, 20, 23]. The perspective we adopt is to study the stochastic Burgers equation by writing its solution as a probabilistic solution [22] plus a term that can be studied with deterministic PDE considerations. One motivation is universality of KPZ and stochastic Burgers equations for a certain class of stochastic PDE growth models, first studied in [28]. For this, we prove universality for SPDEs with general nonlinearities, thereby extending [28, 29], and for many non-stationary initial data, thereby extending [21]. Our perspective lets us also prove explicit rates of convergence to white noise invariant measure of stochastic Burgers for non-stationary initial data, in particular extending the spectral gap result of [23] beyond stationary initial data, though for non-stationary data our convergence will be measured in Wasserstein distance and relative entropy, not via the spectral gap as in [23]. Actually, we extend the spectral gap in [23] to a log-Sobolev inequality. Our methods can also analyze fractional stochastic Burgers equations [23]; we discuss this briefly. Lastly, we note our perspective on the KPZ and stochastic Burgers equations gives a first intrinsic notion of solutions for general continuous initial data, in contrast to Hölder regular data needed for regularity structures [26], paracontrolled distributions [14], and Hölder-regular Brownian bridge data for energy solutions [19, 20].

Uniform spectral gap and orthogeodesic counting for strong convergence of Kleinian groups

Abstract We show convergence of small eigenvalues for geometrically finite hyperbolic n-manifolds under strong limits. For a class of convergent convex sets in a strongly convergent sequence of Kleinian groups, we use the spectral gap of the limit manifold and the exponentially mixing property of the geodesic flow along the strongly convergent sequence to find asymptotically uniform counting formulas for the number of orthogeodesics between the convex sets. In particular, this provides asymptotically uniform counting formulas (with respect to length) for orthogeodesics between converging Margulis tubes, geodesic loops based at converging basepoints, and primitive closed geodesics.

Growth-fragmentation equations with McKendrick–von Foerster boundary condition

The paper concerns the well-posedness and long-term asymptotics of growth–fragmentation equation with unbounded fragmentation rates and McKendrick–von Foerster boundary conditions. We provide three different methods of proving that there is a strongly continuous semigroup solution to the problem and show that it is a compact perturbation of the corresponding semigroup with a homogeneous boundary condition. This allows for transferring the results on the spectral gap available for the later semigroup to the one considered in the paper. We also provide sufficient and necessary conditions for the irreducibility of the semigroup needed to prove that it has asynchronous exponential growth. We conclude the paper by deriving an explicit solution to a special class of growth–fragmentation problems with McKendrick–von Foerster boundary conditions and by finding its Perron eigenpair that determines its long-term behaviour.

The Spectral Gap and Low-Energy Spectrum in Mean-Field Quantum Spin Systems

Abstract A semiclassical analysis based on spin-coherent states is used to establish a classification and novel simple formulae for the spectral gap of mean-field spin Hamiltonians. For gapped systems, we provide a full description of the low-energy spectra based on a second-order approximation to the semiclassical Hamiltonian, hence justifying fluctuation theory at zero temperature for this case. We also point out a shift caused by the spherical geometry in these second-order approximations.

Spectral gaps in a thin-walled infinite rectangular Dirichlet box with a periodic family of cross walls

The Dirichlet spectral problem for the Laplace operator is considered in an infinite thin-walled rectangular box with a periodic family of cross walls whose thickness is proportional to that of the walls. Using asymptotic analysis it is shown that spectral gaps open up in the case of ‘thin’ or ‘sufficiently thick’ cross-walls whose relative thickness is bounded above or below by certain characteristics of model Dirichlet problems in $\mathsf L$- and $\mathsf T$-shaped domains in the plane and in a union of two pairwise orthogonal halves of space layers and a quarter of a space layer. A number of open questions are stated; in particular, because of the lack of information on threshold resonances in the three-dimensional model problem, the structure of the spectrum for cross walls of any intermediate thickness remains unknown. Bibliography: 35 titles.

Spectral gaps in a thin-walled infinite rectangular Dirichlet box with a periodic family of cross walls

Рассматривается спектральная задача Дирихле для оператора Лапласа в бесконечном тонкостенном прямоугольном коробе с периодическим семейством перегородок, толщина которых пропорциональна толщине стенок. При помощи асимптотического анализа доказано раскрытие лакун в спектре при "тонких" или "достаточно толстых" перегородках, для которых относительная толщина ограничена сверху или снизу некоторыми характеристиками модельных задач Дирихле в $\mathsf L$- и $\mathsf T$-образных областях на плоскости и объединении взаимно перпендикулярных двух половин и одной четверти пространственных слоев. Сформулированы многочисленные открытые вопросы, в частности, из-за отсутствия информации о пороговых резонансах в трехмерной модельной задаче осталась неизученной структура спектра в случае перегородок всех промежуточных толщин. Библиография: 35 названий.

Spectral Gap-Based Seismic Survey Design

Seismic imaging in challenging sedimentary basins and reservoirs requires acquiring, processing, and imaging very large volumes of data (tens of terabytes). To reduce the cost of acquisition and the time from acquiring the data to producing a subsurface image, novel acquisition systems based on compressive sensing (CS), low-rank matrix recovery (LRMR), and randomized sampling have been developed and implemented. These approaches allow practitioners to achieve dense wavefield reconstruction from a substantially reduced number of field samples. However, designing acquisition surveys suited for this new sampling paradigm remains a critical and challenging role in oil, gas, and geothermal exploration. Typical random designs studied in the LRMR and CS literature are difficult to achieve by standard industry hardware. For practical purposes, a compromise between stochastic and realizable samples is needed. In this article, we propose a deterministic and computationally cheap tool to alleviate randomized acquisition design, prior to survey deployment and large-scale optimization. We consider universal and deterministic matrix completion results in the context of seismology, where a bipartite graph representation of the source–receiver layout allows for the respective spectral gap (SG) to act as a quality metric for wavefield reconstruction. We provide realistic scenarios to demonstrate the utility of the SG as a flexible tool that can be incorporated into existing survey design workflows for successful seismic data acquisition via low-rank and sparse signal recovery.

Особенности высокоскоростной деформации состаренных сплавов

The sliding of an ensemble of edge dislocations under high-speed deformation (high strain rate deformation) of an aged binary alloy is theoretically analyzed. An analytical expression of the dependence of the dynamic yield strength on the dislocation density is obtained. The conditions under which this dependence is nonmonotonic and has a minimum and maximum are obtained. The minimum occurs during the transition from the dominance of dynamic braking (drag) of dislocations by point defects to the dominance of braking by other dislocations (Taylor hardening). The position of the maximum corresponds to the value of the dislocation density, at which their contribution to the formation of the spectral gap becomes dominant.

Smooth linearization of nonautonomous coupled systems

In a joint work with Palmer we have formulated sufficient conditions under which there exist continuous and invertible transformations of the form $H_n(x,y)$ taking solutions of a coupled system \begin{equation*} x_{n+1} =A_nx_n+f_n(x_n, y_n), \quad y_{n+1}=g_n( y_n), \end{equation*} onto the solutions of the associated partially linearized uncoupled system \begin{equation*} x_{n+1} =A_nx_n, \quad y_{n+1}=g_n( y_n). \end{equation*} In the present work we go one step further and provide conditions under which $H_n$ and $H_n^{-1}$ are smooth in one of the variables $x$ and $y$. We emphasise that our conditions are of a general form and do not involve any kind of dichotomy, nonresonance or spectral gap assumptions for the linear part which are present on most of the related works.

Wasserstein contraction and spectral gap of slice sampling revisited

Wasserstein contraction and spectral gap of slice sampling revisited

Conditioning of linear systems arising from penalty methods

Penalizing incompressibility in the Stokes problem leads, under mildassumptions, to matrices with condition numbers $\kappa =\mathcal{O}(\varepsilon ^{-1}h^{-2})$, with $\varepsilon =$ penalty parameter $\ll1$ and$h= $ meshwidth $<1$. Although $\kappa =\mathcal{O}(\varepsilon^{-1}h^{-2}) $ is large, practical tests seldom report difficulty in solvingthese systems. In the SPD case, using the conjugate gradient method, thisis usually explained by spectral gaps occurring in the penalized coefficientmatrix. Herein we point out a second contributing factor. Since the solutionis approximately incompressible, solution components in the eigenspacesassociated with the penalty terms can be small. As a result, the <i>effective</i> condition number can be much smaller than the standard condition number.

Confining integro-differential equations originating from evolutionary biology: Ground states and long time dynamics

&lt;p style='text-indent:20px;'&gt;We consider nonlinear mutation selection models, known as &lt;i&gt;replicator-mutator&lt;/i&gt; equations in evolutionary biology. They involve a nonlocal mutation kernel and a confining fitness potential. We prove that the long time behaviour of the Cauchy problem is determined by the principal eigenelement of the underlying linear operator. The novelties compared to the literature on these models are about the case of symmetric mutations: we propose a new milder sufficient condition for the existence of a principal eigenfunction, and we provide what is to our knowledge the first quantification of the spectral gap. We also recover existing results in the non-symmetric case, through a new approach.&lt;/p&gt;

Convergence properties of data augmentation algorithms for high-dimensional robit regression

The logistic and probit link functions are the most common choices for regression models with a binary response. However, these choices are not robust to the presence of outliers/unexpected observations. The robit link function, which is equal to the inverse CDF of the Student’s t-distribution, provides a robust alternative to the probit and logistic link functions. A multivariate normal prior for the regression coefficients is the standard choice for Bayesian inference in robit regression models. The resulting posterior density is intractable and a Data Augmentation (DA) Markov chain is used to generate approximate samples from the desired posterior distribution. Establishing geometric ergodicity for this DA Markov chain is important as it provides theoretical guarantees for asymptotic validity of MCMC standard errors for desired posterior expectations/quantiles. Previous work [16] established geometric ergodicity of this robit DA Markov chain assuming (i) the sample size n dominates the number of predictors p, and (ii) an additional constraint which requires the sample size to be bounded above by a fixed constant which depends on the design matrix X. In particular, modern high-dimensional settings where n<p are not considered. In this work, we show that the robit DA Markov chain is trace-class (i.e., the eigenvalues of the corresponding Markov operator are summable) for arbitrary choices of the sample size n, the number of predictors p, the design matrix X, and the prior mean and variance parameters. The trace-class property implies geometric ergodicity. Moreover, this property allows us to conclude that the sandwich robit chain (obtained by inserting an inexpensive extra step in between the two steps of the DA chain) is strictly better than the robit DA chain in an appropriate sense, and enables the use of recent methods to estimate the spectral gap of trace class DA Markov chains.

Exponential ergodicity for a stochastic two-layer quasi-geostrophic model

Ergodic properties of a stochastic medium complexity model for atmosphere and ocean dynamics are analyzed. More specifically, a two-layer quasi-geostrophic model for geophysical flows is studied, with the upper layer being perturbed by additive noise. This model is popular in the geosciences, for instance to study the effects of a stochastic wind forcing on the ocean. A rigorous mathematical analysis however meets with the challenge that in the model under study, the noise configuration is spatially degenerate as the stochastic forcing acts only on the top layer. Exponential convergence of solutions laws to the invariant measure is established, implying a spectral gap of the associated Markov semigroup on a space of Hölder continuous functions. The approach provides a general framework for generalized coupling techniques suitable for applications to dissipative SPDEs. In case of the two-layer quasi-geostrophic model, the results require the second layer to obey a certain passivity condition.

The rigidity of sharp spectral gap in non-negatively curved spaces

We extend the celebrated rigidity of the sharp first spectral gap underRic≥0 to compact infinitesimally Hilbertian spaces with non-negative (weak, also called synthetic) Ricci curvature and bounded (synthetic) dimension i.e. to so-called compact RCD0,N spaces; this is a category of metric measure spaces which in particular includes (Ricci) non-negatively curved Riemannian manifolds, Alexandrov spaces, Ricci limit spaces, Bakry–Émery manifolds along with products, certain quotients and measured Gromov–Hausdorff limits of such spaces. In precise terms, we show in such spaces,λ1=π2diam2if and only if the space is one dimensional with a constant density function. We use new techniques mixing Sobolev theory and singular 1D-localization which might also be of independent interest. As a consequence of the rigidity in the singular setting, we also derive almost rigidity results.

Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups

We provide simple and constructive proofs of Harris-type theorems on the existence and uniqueness of an equilibrium and the speed of equilibration of discrete-time and continuous-time stochastic semigroups. Our results apply both to cases where the relaxation speed is exponential (also called geometric) and to those with no spectral gap, with non-exponential speeds (also called subgeometric). We give constructive estimates in the subgeometric case and discrete-time statements which seem both to be new. The method of proof also differs from previous works, based on semigroup and interpolation arguments, valid for both geometric and subgeometric cases with essentially the same ideas. In particular, we present very simple new proofs of the geometric case.

De Giorgi Argument for Weighted L^2cap L^infty Solutions to the Non-cutoff Boltzmann Equation

This paper gives an affirmative answer to the question of the global existence of Boltzmann equations without angular cutoff in the L^infty -setting. In particular, we show that when the initial data is close to equilibrium and the perturbation is small in L^2 cap L^infty with a polynomial decay tail, the Boltzmann equation has a unique global solution in the weighted L^2cap L^infty -space. In order to overcome the difficulties arising from the singular cross-section and the low regularity, a De Giorgi type argument is crafted in the kinetic context with the help of the averaging lemma. More specifically, we use a strong averaging lemma to obtain suitable L^p-estimates for level-set functions. These estimates are crucial for constructing an appropriate energy functional to carry out the De Giorgi argument. Similar as in Alonso et al. (Rev Mat Iberoam, 2020), we extend local solutions to global ones by using the spectral gap of the linearised Boltzmann operator. The convergence to the equilibrium state is then obtained as a byproduct with relaxations shown in both L^2 and L^infty -spaces.

Explaining the pseudogap through damping and antidamping on the Fermi surface by imaginary spin scattering

The mechanism of the pseudogap observed in hole-doped cuprates remains one of the central puzzles in condensed matter physics. We analyze this phenomenon via a Feynman-diagrammatic inspection of the Hubbard model. Our approach captures the pivotal interplay between Mott localization and Fermi surface topology beyond weak-coupling spin fluctuations, which would open a spectral gap near hot spots. We show that strong coupling and particle-hole asymmetry trigger a very different mechanism: a large imaginary part of the spin-fermion vertex promotes damping of antinodal fermions and, at the same time, protects the nodal Fermi arcs (antidamping). Our analysis naturally explains puzzling features of the pseudogap observed in experiments, such as Fermi arcs being cut off at the antiferromagnetic zone boundary and the subordinate role of hot spots.

Aldous’ spectral gap property for normal Cayley graphs on symmetric groups

Aldous’ spectral gap conjecture states that the second largest eigenvalue of any connected Cayley graph on the symmetric group Sn with respect to a set of transpositions is achieved by the standard representation of Sn. This celebrated conjecture, which was proved in its general form in 2010, has inspired much interest in searching for other families of Cayley graphs on Sn with the property that the largest eigenvalue strictly smaller than the degree is attained by the standard representation of Sn. In this paper, we prove three results on normal Cayley graphs on Sn possessing this property for sufficiently large n, one of which can be viewed as a generalization of the “normal” case of Aldous’ spectral gap conjecture.