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Coupling Weyl quasiparticles and charge density waves (CDWs) can lead to fascinating band renormalization and many-body effects beyond band folding and Peierls gaps. For the quasi-one-dimensional chiral compound (TaSe4)2I with an incommensurate CDW transition at TC = 263 K, photoemission mappings thus far are intriguing due to suppressed emission near the Fermi level. Models for this unconventional behavior include axion insulator phases, correlation pseudogaps, polaron subbands, bipolaron bound states, etc. Our photoemission measurements show sharp quasiparticle bands crossing the Fermi level at T > TC, but for T < TC, these bands retain their dispersions with no Peierls or axion gaps at the Weyl points. Instead, occupied band edges recede from the Fermi level, opening a spectral gap. Our results confirm localization of quasiparticles (holes created by photoemission) is the key physics, which suppresses spectral weights over an energy window governed by incommensurate modulation and inherent phase defects of CDW.

We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for polytopes defined by integer constraints in terms of the height of the integers and certain subdeterminants of the constraint matrix, which in some cases improves previously known results. The second is a smoothed analysis bound: given an appropriately normalized polytope, we add small Gaussian noise to each constraint. We consider a natural geometric measure on the vertices of the perturbed polytope (corresponding to the mean curvature measure of its polar) and show that with high probability there exists a "giant component" of vertices, with measure 1 - o (1) and polynomial diameter. Both bounds rely on spectral gaps—of a certain Schrödinger operator in the first case, and a certain continuous time Markov chain in the second—which arise from the log-concavity of the volume of a simple polytope in terms of its slack variables. [ABSTRACT FROM AUTHOR]

We study the effect of non-negative potentials on the spectral gap of one-dimensional Schrödinger operators in the limit of large intervals. We derive upper bounds on the gap for different classes of potentials and show, as a main result, that the spectral gap of a Schrödinger operator with a non-zero and sufficiently fast decaying potential closes strictly faster than the gap of the free Laplacian. We show optimality of this result in some sense and establish a conjecture towards the actual decay rate of the spectral gap. [ABSTRACT FROM AUTHOR]

It is known that the simple slice sampler has robust convergence properties; however, the class of problems where it can be implemented is limited. In contrast, we consider hybrid slice samplers which are easily implementable and where another Markov chain approximately samples the uniform distribution on each slice. Under appropriate assumptions on the Markov chain on the slice, we give a lower bound and an upper bound of the spectral gap of the hybrid slice sampler in terms of the spectral gap of the simple slice sampler. An immediate consequence of this is that the spectral gap and geometric ergodicity of the hybrid slice sampler can be concluded from the spectral gap and geometric ergodicity of the simple version, which is very well understood. These results indicate that robustness properties of the simple slice sampler are inherited by (appropriately designed) easily implementable hybrid versions. We apply the developed theory and analyze a number of specific algorithms, such as the stepping-out shrinkage slice sampling, hit-and-run slice sampling on a class of multivariate targets, and an easily implementable combination of both procedures on multidimensional bimodal densities. [ABSTRACT FROM AUTHOR]

Let a particle start at some point in the unit interval I:=[0,1]$$ I:= \left[0,1\right] $$ and undergo Brownian motion in I$$ I $$ until it hits one of the end points. At this instant, the particle stays put for a finite holding time with an exponential distribution and then jumps back to a point inside I$$ I $$ with a probability density μ0$$ {\mu}_0 $$ or μ1$$ {\mu}_1 $$ parametrized by the boundary point it was at. The process starts afresh. The same evolution repeats independently each time. Many probabilistic aspects of this diffusion process are investigated in the paper by Peng and Li listed in the reference. The authors in the cited paper call this process diffusion with holding and jumping (DHJ). Our simple aim in this paper is to analyze the eigenvalues of a nonlocal boundary problem arising from this process. In particular, we answer a question on the spectral gap of the DHJ process raised at the end of the paper by Peng and Li. [ABSTRACT FROM AUTHOR]

AbstractThis article studies the convergence properties of trans-dimensional MCMC algorithms when the total number of models is finite. It is shown that, for reversible and some non-reversible trans-dimensional Markov chains, under mild conditions, geometric convergence is guaranteed if the Markov chains associated with the within-model moves are geometrically ergodic. This result is proved in an L2 framework using the technique of Markov chain decomposition. While the technique was previously developed for reversible chains, this work extends it to the point that it can be applied to some commonly used non-reversible chains. The theory herein is applied to reversible jump algorithms for three Bayesian models: a probit regression with variable selection, a Gaussian mixture model with unknown number of components, and an autoregression with Laplace errors and unknown model order. [ABSTRACT FROM AUTHOR]

We establish the existence of a spectral gap for the transfer operator induced on by a generic holomorphic endomorphism and a suitable continuous weight and its perturbations on various functional spaces, which is new even in dimension one. Thanks to the spectral gap, we establish an exponential speed of convergence for the equidistribution of the backward orbits of points towards the conformal measure and the exponential mixing. Moreover, as an immediate consequence, we obtain a full list of statistical properties for the equilibrium states: CLT, Berry-Esseen Theorem, local CLT, ASIP, LIL, LDP, almost sure CLT. Many of these properties are new even in dimension one, some even in the case of zero weight function (i.e., for the measure of maximal entropy). [ABSTRACT FROM AUTHOR]

Understanding efficient modifications to improve network functionality is a fundamental problem of scientific and industrial interest. We study the response of network dynamics against link modifications on a weakly connected directed graph consisting of two strongly connected components: an undirected star and an undirected cycle. We assume that there are directed edges starting from the cycle and ending at the star (master–slave formalism). We modify the graph by adding directed edges of arbitrarily large weights starting from the star and ending at the cycle (opposite direction of the cutset). We provide criteria (based on the sizes of the star and cycle, the coupling structure, and the weights of cutset and modification edges) that determine how the modification affects the spectral gap of the Laplacian matrix. We apply our approach to understand the modifications that either enhance or hinder synchronization in networks of chaotic Lorenz systems as well as Rössler. Our results show that the hindrance of collective dynamics due to link additions is not atypical as previously anticipated by modification analysis and thus allows for better control of collective properties. [ABSTRACT FROM AUTHOR]

We propose the multivariate locally stationary wavelet (mvLSW) process to analyze surface turbulent fluxes in nonstationary atmospheric conditions. Using theoretical spectral characteristics, we generated synthetic data representing stationary and nonstationary turbulence time series. This data enables us to explore the impact of mesoscale atmospheric flows on the stationary microscale turbulence field and detect the spectral gap in the time-varying cospectra. Applying this approach to experimental data collected in Fairbanks, Alaska and Bogota, Colombia, we demonstrated the ability to detect cospectral gaps and compute bandwidth-limited turbulent fluxes arising from stationary components of the atmospheric flow. These findings underscore the importance of considering scale-dependent atmospheric forcing when comparing model and experimental data. [ABSTRACT FROM AUTHOR]

Aldous and Fill (2002) conjectured that the maximum relaxation time for the random walk on a connected regular graph with n vertices is (1 + o (1)) 3 n 2 2 π 2 . A conjecture by Guiduli and Mohar (1996) predicts the structure of graphs whose algebraic connectivity μ is the smallest among all connected graphs whose minimum degree δ is a given d. We prove that this conjecture implies the Aldous–Fill conjecture for odd d. We pose another conjecture on the structure of d -regular graphs with minimum μ , and show that this also implies the Aldous–Fill conjecture for even d. In the literature, it has been noted empirically that graphs with small μ tend to have a large diameter. In this regard, Guiduli (1996) asked if the cubic graphs with maximum diameter have algebraic connectivity smaller than all others. Motivated by these, we investigate the interplay between the graphs with maximum diameter and those with minimum algebraic connectivity. We show that the answer to Guiduli problem in its general form, that is for d -regular graphs for every d ≥ 3 is negative. We aim to develop an asymptotic formulation of the problem. It is proven that d -regular graphs for d ≥ 5 as well as graphs with δ = d for d ≥ 4 with asymptotically maximum diameter, do not necessarily exhibit the asymptotically smallest μ. We conjecture that d -regular graphs (or graphs with δ = d) that have asymptotically smallest μ , should have asymptotically maximum diameter. The above results rely heavily on our understanding of the structure as well as optimal estimation of the algebraic connectivity of nearly maximum-diameter graphs, from which the Aldous–Fill conjecture for this family of graphs also follows. [ABSTRACT FROM AUTHOR]

Covariate shift in regression problems and the associated distribution mismatch between training and test data is a commonly encountered phenomenon in machine learning. In this paper, we extend recent results on nonparametric convergence rates for i.i.d. data to Markovian dependence structures. We demonstrate that under Hölder smoothness assumptions on the regression function, convergence rates for the generalization risk of a Nadaraya–Watson kernel estimator are determined by the similarity between the invariant distributions associated to source and target Markov chains. The similarity is explicitly captured in terms of a bandwidth-dependent similarity measure recently introduced in Pathak, Ma and Wainwright [ ICML , 2022]. Precise convergence rates are derived for the particular cases of finite Markov chains and spectral gap Markov chains for which the similarity measure between their invariant distributions grows polynomially with decreasing bandwidth. For the latter, we extend the notion of a distribution transfer exponent from Kpotufe and Martinet [ Ann. Stat. , 49(6), 2021] to kernel transfer exponents of uniformly ergodic Markov chains in order to generate a rich class of Markov kernel pairs for which convergence guarantees for the covariate shift problem can be formulated. [ABSTRACT FROM AUTHOR]

We investigate three aspects of weak* convergence of the n-step distributions of random walks on finite volume homogeneous spaces $ G/\Gamma $ G / Γ of semisimple real Lie groups. First, we look into the obvious obstruction to the upgrade from Cesàro to non-averaged convergence: periodicity. We give examples where it occurs and conditions under which it does not. In a second part, we prove convergence towards Haar measure with exponential speed from almost every starting point. Finally, we establish a strong uniformity property for the Cesàro convergence towards Haar measure for uniquely ergodic random walks. [ABSTRACT FROM AUTHOR]

We study the geometry of compact geodesic spaces with trivial first Betti number admitting large finite groups of isometries. We show that if a finite group G acts by isometries on a compact geodesic space X whose first Betti number vanishes, then diam (X) / diam (X / G) ≤ 4 | G | . For a group G and a finite symmetric generating set S, P k (Γ (G , S)) denotes the 2-dimensional CW-complex whose 1-skeleton is the Cayley graph Γ of G with respect to S and whose 2-cells are m-gons for 0 ≤ m ≤ k , defined by the simple graph loops of length m in Γ , up to cyclic permutations. Let G be a finite abelian group with | G | ≥ 3 and S a symmetric set of generators for which P k (Γ (G , S)) has trivial first Betti number. We show that the first nontrivial eigenvalue - λ 1 of the Laplacian on the Cayley graph satisfies λ 1 ≥ 2 - 2 cos (2 π / k) . We also give an explicit upper bound on the diameter of the Cayley graph of G with respect to S of the form O (k 2 | S | log | G |) . Related explicit bounds for the Cheeger constant and Kazhdan constant of the pair (G, S) are also obtained. [ABSTRACT FROM AUTHOR]

We treat the linear heat equation in a periodic waveguide Π ⊂ R d , with a regular enough boundary, by using the Floquet transform methods. Applying the Floquet transform F to the equation yields a heat equation with mixed boundary conditions on the periodic cell ϖ of Π , and we analyse the connection between the solutions of the two problems. The considerations involve a description of the spectral projections onto subspaces H S ⊂ L 2 (Π) corresponding certain spectral components. We also show that the translated Wannier functions form an orthonormal basis in H S . [ABSTRACT FROM AUTHOR]

In this paper, we propose new Metropolis–Hastings and simulated annealing algorithms on a finite state space via modifying the energy landscape. The core idea of landscape modification rests on introducing a parameter c , such that the landscape is modified once the algorithm is above this threshold parameter to encourage exploration, while the original landscape is utilized when the algorithm is below the threshold for exploitation purposes. We illustrate the power and benefits of landscape modification by investigating its effect on the classical Curie–Weiss model with Glauber dynamics and external magnetic field in the subcritical regime. This leads to a landscape-modified mean-field equation, and with appropriate choice of c the free energy landscape can be transformed from a double-well into a single-well landscape, while the location of the global minimum is preserved on the modified landscape. Consequently, running algorithms on the modified landscape can improve the convergence to the ground state in the Curie–Weiss model. In the setting of simulated annealing, we demonstrate that landscape modification can yield improved or even subexponential mean tunnelling time between global minima in the low-temperature regime by appropriate choice of c , and we give a convergence guarantee using an improved logarithmic cooling schedule with reduced critical height. We also discuss connections between landscape modification and other acceleration techniques, such as Catoni's energy transformation algorithm, preconditioning, importance sampling, and quantum annealing. The technique developed in this paper is not limited to simulated annealing, but is broadly applicable to any difference-based discrete optimization algorithm by a change of landscape. [ABSTRACT FROM AUTHOR]

We study the simple random walk on trees and give estimates on the mixing and relaxation times. Relying on a seminal result by Basu, Hermon and Peres characterizing cutoff on trees, we give geometric criteria that are easy to verify and allow to determine whether the cutoff phenomenon occurs. We provide a general characterization of families of trees with cutoff, and show how our criteria can be used to prove the absence of cutoff for several classes of trees, including spherically symmetric trees, Galton–Watson trees of a fixed height, and sequences of random trees converging to the Brownian continuum random tree. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We introduce a natural notion of mean (or average) distance in the context of compact metric graphs, and study its relation to geometric properties of the graph. We show that it exhibits a striking number of parallels to the reciprocal of the spectral gap of the graph Laplacian with standard vertex conditions: it is maximised among all graphs of fixed length by the path graph (interval), or by the loop in the restricted class of doubly connected graphs, and it is minimised among all graphs of fixed length and number of edges by the equilateral flower graph. We also establish bounds for the correctly scaled product of the spectral gap and the square of the mean distance which depend only on combinatorial, and not metric, features of the graph. This raises the open question whether this product admits absolute upper and lower bounds valid on all compact metric graphs. [ABSTRACT FROM AUTHOR]

In this paper, we study the discrete nonlinear Schrödinger equation - Δ u + V (x) - ρ (| x | 2 + 1) u = f (x , u) , u ∈ ℓ 2 (Z N) , where N ≥ 3 , V is a bounded periodic potential and 0 lies in a spectral gap of the Schrödinger operator - Δ + V . The resulting problem engages two major difficulties: one is that the associated functional is strongly indefinite and the other is the lack of compactness of the Cerami sequence. We overcome these two major difficulties by the generalized linking theorem and Lions lemma. This enables us to establish the existence and asymptotic behavior of ground state solutions for small ρ ≥ 0 . [ABSTRACT FROM AUTHOR]

We consider a class of endomorphisms that contains a set of piecewise partially hyperbolic dynamics semi-conjugated to non-uniformly expanding maps. Our goal is to study a class of endomorphisms that preserve a foliation that is almost everywhere uniformly contracted, with possible discontinuity sets parallel to the contracting direction. We apply the spectral gap property and the $ \zeta $-Hölder regularity of the disintegration of its equilibrium states to prove a quantitative statistical stability statement. More precisely, under deterministic perturbations of the system of size $ \delta $, we show that the $ F $-invariant measure varies continuously with respect to a suitable anisotropic norm. Furthermore, we establish that certain interesting classes of perturbations exhibit a modulus of continuity estimated by $ D_2\delta^\zeta \log \delta $, where $ D_2 $ is a constant. [ABSTRACT FROM AUTHOR]

We estimate the lowest eigenvalue in the gap of the essential spectrum of a Dirac operator with mass in terms of a Lebesgue norm of the potential. Such a bound is the counterpart for Dirac operators of the Keller estimates for the Schrödinger operator, which are equivalent to some Gagliardo-Nirenberg-Sobolev interpolation inequalities. Domain, self-adjointness, optimality and critical values of the norms are addressed, while the optimal potential is given by a Dirac equation with a Kerr nonlinearity. A new critical bound appears, which is the smallest value of the norm of the potential for which eigenvalues may reach the bottom of the gap in the essential spectrum. The Keller estimate is then extended to a Lieb-Thirring inequality for the eigenvalues in the gap. Most of our result are established in the Birman-Schwinger reformulation. [ABSTRACT FROM AUTHOR]

We investigate relative property (T) of Kazhdan-Margulis for (SL 2 (R) ⋉ R n , R n) , mainly where R is a discrete finitely generated commutative ring. The action of SL 2 (R) on R n is defined through admissible lattices of the irreducible representations of sl 2 (C). Both positive and negative results are obtained. Applications of these results will be given, including the establishment of property (T) for certain Kac-Moody type groups over commutative rings. [ABSTRACT FROM AUTHOR]

We establish a tunneling formula for a Schrödinger operator with symmetric double-well potential and homogeneous magnetic field, in dimension two. Each well is assumed to be radially symmetric and compactly supported. We obtain an asymptotic formula for the difference between the two first eigenvalues of this operator, that is exponentially small in the semiclassical limit. [ABSTRACT FROM AUTHOR]

Polar slice sampling, a Markov chain construction for approximate sampling, performs, under suitable assumptions on the target and initial distribution, provably independent of the state space dimension. We extend the aforementioned result of Roberts and Rosenthal (Stoch Model 18(2):257–280, 2002) by developing a theory which identifies conditions, in terms of a generalized level set function, that imply an explicit lower bound on the spectral gap even in a general slice sampling context. Verifying the identified conditions for polar slice sampling yields a lower bound of 1/2 on the spectral gap for arbitrary dimension if the target density is rotationally invariant, log-concave along rays emanating from the origin and sufficiently smooth. The general theoretical result is potentially applicable beyond the polar slice sampling framework. [ABSTRACT FROM AUTHOR]

For a single matrix (operator) it is well-known that the spectral gap is an important quantity, as well as its estimate and computation. Here we consider, for the first time in the literature, the computation of its extension to a finite family of matrices, in other words the difference between the joint spectral radius (in short JSR, which we call here the first Lyapunov exponent) and the second Lyapunov exponent (denoted as SLE). The knowledge of joint spectral characteristics and of the spectral gap of a family of matrices is important in several applications, as in the analysis of the regularity of wavelets, multiplicative matrix semigroups and the convergence speed in consensus algorithms. As far as we know the methods we propose are the first able to compute this quantity to any given accuracy. For computation of the spectral gap one needs first to compute the JSR. A popular tool that is used to this purpose is the invariant polytope algorithm, which relies on the finiteness property of the family of matrices, when this holds true. In this paper we show that the SLE may not possess the finiteness property, although it can be efficiently approximated with an arbitrary precision. The corresponding algorithm and two effective estimates are presented. Moreover, we prove that the SLE possesses a weak finiteness property, whenever the leading eigenvalue of the dominant product is real. This allows us to find in certain situations the precise value of the SLE. Numerical results are demonstrated along with applications in the theory of multiplicative matrix semigroups and in the wavelets theory. [ABSTRACT FROM AUTHOR]

Hydrocarbons are one of the subclasses of organic compounds that comprise exactly of hydrogen and carbon. Alkanes are one of the types of hydrocarbons that have chemical formula CnH2n+2. Isomers are molecules with identical chemical formula but different structural arrangements, leading to variations in their spectral properties as their corresponding molecular graphs also differ in structure. This exploration is motivated by the understanding that variations in structural configurations manifest as differences in spectral properties, as evidenced by alterations in their respective molecular graphs. Alkanes with ten carbon atoms are called decanes. Our study employs a multifaceted approach, encompassing the determination of spectral properties and the calculation of eigenvalue-based entropy for the C10H22 decane isomers. This analysis is undertaken with the goal of unravelling the intricate relationships between structural variations and corresponding spectral bounds. Notably, our investigation extends beyond the realm of molecular structures to draw connections with physico-chemical properties. Through meticulous comparison of the obtained spectral data with the known characteristics of C10H22 isomers, we unveil interesting correlations among the characteristics. We establish that the spectral gap, a key parameter in our study, intriguingly exhibits a maximal correlation with the refractive index of the isomers. This finding not only enhances our understanding of the spectral intricacies of decane isomers but also underscores the practical implications of such research. The correlation between spectral gap and refractive index opens avenues for predicting and manipulating the optical properties of hydrocarbons, offering potential applications in diverse fields, from materials science to optics. In essence, this study bridges the gap between molecular structure and macroscopic properties, shedding light on the intricate interplay between isomeric variations and their consequential effects on spectral characteristics.

We consider the spatially-dependent diffusion tensors that maximize the rate of convergence to equilibrium of solutions to the heat equation in inhomogeneous media. We formulate a variational problem for the optimal spectral gap, subject to an Lp constraint on the diffusion tensors, and solve a relaxed version of this variational problem. This provides an upper bound for the optimal spectral gap. The solution is characterized in terms of the extremals of Sobolev and Poincaré type inequalities. To illustrate this result, we give a presentation of the one-dimensional case. In this case, we can adapt the known closed solution of a related eigenvalue problem to write the solution to the relaxed variational problem, arriving at an expression for the upper bound on the optimal spectral gap. The sharpness of this upper bound is then investigated by comparison to numerical simulations. [ABSTRACT FROM AUTHOR]

The classical Fatou theorem identifies bounded harmonic functions on the unit disk with bounded measurable functions on the boundary circle. We extend this theorem to bounded harmonic maps. [ABSTRACT FROM AUTHOR]

The spectral gap of the Neumann and Dirichlet Laplacians are each known to have a sharp positive lower bound among convex domains of a given diameter. Between these cases, for each positive value of the Robin parameter, an analogous sharp lower bound on the spectral gap is conjectured. In this paper, we show that the extension of this conjecture to negative Robin parameters fails by proving that the spectral gap of double cone domains are exponentially small, for each fixed parameter value. [ABSTRACT FROM AUTHOR]

We bound the second eigenvalue of random d$$ d $$‐regular graphs, for a wide range of degrees d$$ d $$, using a novel approach based on Fourier analysis. Let Gn,d$$ {G}_{n,d} $$ be a uniform random d$$ d $$‐regular graph on n$$ n $$ vertices, and λ(Gn,d)$$ \lambda \left({G}_{n,d}\right) $$ be its second largest eigenvalue by absolute value. For some constant c>0$$ c>0 $$ and any degree d$$ d $$ with log10n≪d≤cn$$ {\log}^{10}n\ll d\le cn $$, we show that λ(Gn,d)=(2+o(1))d(n−d)/n$$ \lambda \left({G}_{n,d}\right)=\left(2+o(1)\right)\sqrt{d\left(n-d\right)/n} $$ asymptotically almost surely. Combined with earlier results that cover the case of sparse random graphs, this fully determines the asymptotic value of λ(Gn,d)$$ \lambda \left({G}_{n,d}\right) $$ for all d≤cn$$ d\le cn $$. To achieve this, we introduce new methods that use mechanisms from discrete Fourier analysis, and combine them with existing tools and estimates on d$$ d $$‐regular random graphs—especially those of Liebenau and Wormald. [ABSTRACT FROM AUTHOR]

Turbulent motions in the Arctic stable boundary layer are characterized by intermittency, but they are rarely investigated due to limited observations, in particular over the sea‐ice surface. In the present study, we explore the characteristics of turbulent intermittency over the Arctic sea‐ice surface using data collected during the Multidisciplinary drifting Observation for the Study of Arctic Climate expedition from October 2019 to September 2020. We first develop a new algorithm, which performs well in identifying the spectral gap over the Arctic sea‐ice surface. Then the characteristics of intermittency are investigated. It is found that the strength of intermittency increases under the conditions of light surface wind speed, small surface wind speed gradient, and strong surface air temperature gradient. The momentum flux, sensible heat flux, and latent heat flux calculated by raw eddy‐covariance fluctuations are overestimated by 3%, 10%, and 24%, respectively, because submesoscale motions are included. Furthermore, the characteristics of the atmospheric boundary layer structure under various intermittency conditions reveal that strong low‐level jets are favorable to surface turbulent motions that result in weak intermittency, while strong temperature inversions above the surface layer suppress surface turbulent motions and lead to strong intermittency. Plain Language Summary: In polar regions, turbulent mixing in the near‐surface layer is weak and is often modulated by strong stable stratification, thus the intermittent state of turbulence is universal. The intermittency might lead to inaccuracy of the Monin‐Obukhov similarity theory, which assumes that turbulence is fully developed. In this study, by using a year of eddy‐covariance observations collected over the Arctic sea‐ice surface, we first develop an automatic algorithm to identify the spectral gap where the Hilbert spectrum presents an abrupt drop in low frequency. Using only high‐frequency turbulent motions, the turbulent data is reconstructed. This algorithm has the potential to quantify the characteristics of turbulence intermittency in polar regions. Based on our analysis, the intermittency results in the greatest relative influence on the latent heat flux compared with the momentum and sensible heat fluxes. Furthermore, we found the following meteorological conditions often lead to the enhancement of intermittency: low surface wind speed and wind speed gradient; strong surface air temperature gradient; weak low‐level jets; and strong temperature inversions above the surface layer. These findings help to improve our understanding of turbulent intermittency over the Arctic sea‐ice surface. Key Points: A new algorithm, which performs well in identifying the spectral gap over the Arctic sea‐ice surface, is proposedThe characteristics of intermittency under different meteorological conditions are revealedThe effects of intermittency on eddy‐covariance fluxes are assessed [ABSTRACT FROM AUTHOR]

We obtain a fundamental gap estimate for classes of bounded domains with quantitative control on the boundary in a complete manifold with integral bounds on the negative part of the Ricci curvature. This extends the result of Oden, Sung, and Wang [Trans. Amer. Math. Soc. 351 (1999), no. 9, 3533–3548] to Lp$L^p$‐Ricci curvature assumptions, p>n/2$p>n/2$. To achieve our result, it is shown that the domains under consideration are John domains, what enables us to obtain an estimate on the first nonzero Neumann eigenvalue, which is of independent interest. [ABSTRACT FROM AUTHOR]

We obtain various new limit theorems for random walks on SL 2 (C) under low moment conditions. For non-elementary measures with a finite second moment we prove a Local Limit Theorem for the norm cocycle, yielding the optimal version of a theorem of É. Le Page. For measures with a finite third moment, we obtain the Local Limit Theorem for the matrix coefficients, improving a recent result of Grama-Quint-Xiao and the authors, and Berry–Esseen bounds with optimal rate O (1 / n) for the norm cocycle and the matrix coefficients. The main tool is a detailed study of the spectral properties of the Markov operator and its purely imaginary perturbations acting on different function spaces. We introduce, in particular, a new function space derived from the Sobolev space W 1 , 2 that provides uniform estimates. [ABSTRACT FROM AUTHOR]

We consider the spectral gap of a uniformly chosen random (d 1 , d 2) -biregular bipartite graph G with | V 1 | = n , | V 2 | = m , where d 1 , d 2 could possibly grow with n and m. Let A be the adjacency matrix of G. Under the assumption that d 1 ≥ d 2 and d 2 = O (n 2 / 3) , we show that λ 2 (A) = O (d 1) with high probability. As a corollary, combining the results from (Tikhomirov and Yousse in Ann Probab 47(1):362–419, 2019), we show that the second singular value of a uniform random d-regular digraph is O (d) for 1 ≤ d ≤ n / 2 with high probability. Assuming d 2 is fixed and d 1 = O (n 2) , we further prove that for a random (d 1 , d 2) -biregular bipartite graph, | λ i 2 (A) - d 1 | = O (d 1) for all 2 ≤ i ≤ n + m - 1 with high probability. The proofs of the two results are based on the size biased coupling method introduced in Cook et al. (Ann Probab 46(1):72–125, 2018) for random d-regular graphs and several new switching operations we define for random bipartite biregular graphs. [ABSTRACT FROM AUTHOR]

In this paper, we investigate via an asymptotic method the opening of gaps in the spectrum of a stiff problem for the Laplace operator − Δ in R 2 perforated by contiguous circular holes. The density and the stiffness constants are of order ε − 2 m and ε − 1 in the holes with m ∈ (0 , 1 / 2). We provide an explicit expression of the leading terms of the eigenvalues and the corresponding eigenfunctions which are related to the Bessel functions of the first kind. [ABSTRACT FROM AUTHOR]

Recently, great attention has been devoted to the problem of the undecidability of specific questions in quantum mechanics. In this context, it has been shown that the problem of the existence of a spectral gap, i.e., energy difference between the ground state and the first excited state, is algorithmically undecidable. Using this result herein proves that the existence of a quantum phase transition, as inferred from specific microscopic approaches, is an undecidable problem, too. Indeed, some methods, usually adopted to study quantum phase transitions, rely on the existence of a spectral gap. Since there exists no algorithm to determine whether an arbitrary quantum model is gapped or gapless, and there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics, it infers that the existence of quantum phase transitions is an undecidable problem.

We study the asymptotic behaviour of the spectral gap of Schrödinger operators in two and higher dimensions and in a limit where the volume of the domain tends to infinity. Depending on properties of the underlying potential, we will find different asymptotic behaviours of the gap. In some cases the gap behaves as the gap of the free Dirichlet Laplacian and in some cases it does not. [ABSTRACT FROM AUTHOR]

We provide a scheme of a recent stability result for a family of Gagliardo-Nirenberg-Sobolev (GNS) inequalities, which is equivalent to an improved entropy – entropy production inequality associated with an appropriate fast diffusion equation (FDE) written in self-similar variables. This result can be rephrased as an improved decay rate of the entropy of the solution of (FDE) for well prepared initial data. There is a family of Caffarelli-Kohn-Nirenberg (CKN) inequalities which has a very similar structure. When the exponents are in a range for which the optimal functions for (CKN) are radially symmetric, we investigate how the methods for (GNS) can be extended to (CKN). In particular, we prove that the solutions of the evolution equation associated to (CKN) also satisfy an improved decay rate of the entropy, after an explicit delay. However, the improved rate is obtained without assuming that initial data are well prepared, which is a major difference with the (GNS) case. [ABSTRACT FROM AUTHOR]

This paper gives an affirmative answer to the question of the global existence of Boltzmann equations without angular cutoff in the L ∞ -setting. In particular, we show that when the initial data is close to equilibrium and the perturbation is small in L 2 ∩ L ∞ with a polynomial decay tail, the Boltzmann equation has a unique global solution in the weighted L 2 ∩ L ∞ -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 ∞ -spaces. [ABSTRACT FROM AUTHOR]

Aldous and Fill conjectured that the maximum relaxation time for the random walk on a connected regular graph with n $n$ vertices is (1+o(1))3n22π2 $(1+o(1))\frac{3{n}^{2}}{2{\pi }^{2}}$. This conjecture can be rephrased in terms of the spectral gap as follows: the spectral gap (algebraic connectivity) of a connected k $k$‐regular graph on n $n$ vertices is at least (1+o(1))2kπ23n2 $(1+o(1))\frac{2k{\pi }^{2}}{3{n}^{2}}$, and the bound is attained for at least one value of k $k$. We determine the structure of connected quartic graphs on n $n$ vertices with a minimum spectral gap which enables us to show that the minimum spectral gap of connected quartic graphs on n $n$ vertices is (1+o(1))4π2n2 $(1+o(1))\frac{4{\pi }^{2}}{{n}^{2}}$. From this result, the Aldous–Fill conjecture follows for k=4 $k=4$. [ABSTRACT FROM AUTHOR]

In data-parallel optimization of machine learning models, workers collaborate to improve their estimates of the model: more accurate gradients allow them to use larger learning rates and optimize faster. In the decentralized setting, in which workers communicate over a sparse graph, current theory fails to capture important aspects of real-world behavior. First, the 'spectral gap' of the communication graph is not predictive of its empirical performance in (deep) learning. Second, current theory does not explain that collaboration enables larger learning rates than training alone. In fact, it prescribes smaller learning rates, which further decrease as graphs become larger, failing to explain convergence dynamics in infinite graphs. This paper aims to paint an accurate picture of sparsely-connected distributed optimization. We quantify how the graph topology influences convergence in a quadratic toy problem and provide theoretical results for general smooth and (strongly) convex objectives. Our theory matches empirical observations in deep learning, and accurately describes the relative merits of different graph topologies. This paper is an extension of the conference paper by Vogels et al. (2022). Code: github.com/epfml/topology-in-decentralized-learning. [ABSTRACT FROM AUTHOR]

We consider the linear water-wave problem in a periodic channel Π h ⊂ R 3 , which is shallow except for a periodic array of deep potholes in it. Motivated by applications to surface wave propagation phenomena, we study the band-gap structure of the essential spectrum in the linear water-wave system, which includes the spectral Steklov boundary condition posed on the free water surface. We apply methods of asymptotic analysis, where the most involved step is the construction and analysis of an appropriate boundary layer in a neighborhood of the joint of the potholes with the thin part of the channel. Consequently, the existence of a spectral gap for small enough h is proven. [ABSTRACT FROM AUTHOR]

We analyse the L 2 Hilbert complexes naturally associated to a non-compact complex manifold, namely the ones which originate from the Dolbeault and the Aeppli-Bott-Chern complexes. In particular we define the L 2 Aeppli-Bott-Chern Hilbert complex and examine its main properties on general Hermitian manifolds, on complete Kähler manifolds and on Galois coverings of compact complex manifolds. The main results are achieved through the study of self-adjoint extensions of various differential operators whose kernels, on compact Hermitian manifolds, are isomorphic to either Aeppli or Bott-Chern cohomology. [ABSTRACT FROM AUTHOR]