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Sampling Gibbs measures at low temperatures is an important but computationally challenging task. Numerical evidence suggests that the infinite-swapping algorithm (isa) is a promising method. The isa can be seen as an improvement of the parallel tempering replica method. We rigorously analyze the ergodic properties of the isa in the low temperature regime, deducing asymptotic estimates for the spectral gap (or Poincaré constant), optimal in dimension one, and an estimate for the log-Sobolev constant. Our main results indicate that the effective energy barrier can be reduced drastically using the isa compared to the classical over-damped Langevin dynamics. As a corollary, we derive a concentration inequality showing that sampling is also improved by an exponential factor. Finally, we study simulated annealing for the isa and prove that the isa again outperforms the over-damped Langevin dynamics. [ABSTRACT FROM AUTHOR]

Langevin diffusion (LD) is one of the main workhorses for sampling problems. However, its convergence rate can be significantly reduced if the target distribution is a mixture of multiple densities, especially when each component density concentrates around a different mode. Replica exchange Langevin diffusion (ReLD) is a sampling method that can circumvent this issue. This approach can be further extended to multiple replica exchange Langevin diffusion (mReLD). While ReLD and mReLD have been used extensively in statistics, molecular dynamics, and other applications, there is limited existing analysis on its convergence rate and choices of the temperatures. This paper addresses these problems assuming the target distribution is a mixture of log-concave densities. We show ReLD can obtain constant or better convergence rates. We also show mReLD with K additional LDs can achieve the same results while the exchange frequency only needs to be (1 / K) -th power of the one in ReLD. [ABSTRACT FROM AUTHOR]

We construct a generalisation of Bakry–Émery 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. [ABSTRACT FROM AUTHOR]

We consider the Fokker-Planck equation with subcritical confinement force field which may not derive from a potential function. We prove the existence of a unique positive equilibrium of mass one and we establish some subgeometric, or geometric, rate of convergence to a multiple of this equilibrium (depending on the space to which belongs the initial datum) in many spaces. Our results generalize similar results introduced by Toscani, Villani [33] and Röckner, Wang [31] for some forces associated to a potential and extended by Douc, Fort, Guillin [12] and Bakry, Cattiaux, Guillin [4] for some general forces: however in our approach the spaces are more general, and the rates of convergence to equilibrium are sharper. [ABSTRACT FROM AUTHOR]

Let (g n) n ⩾ 1 be a sequence of independent identically distributed d × d real random matrices with Lyapunov exponent λ. For any starting point x on the unit sphere in R d , we deal with the norm | G n x | , where G n ≔ g n ... g 1 . The goal of this paper is to establish precise asymptotics for large deviation probabilities P (log | G n x | ⩾ n (q + l)) , where q > λ is fixed and l is vanishing as n → ∞. We study both invertible matrices and positive matrices and give analogous results for the couple (X n x , log | G n x |) with target functions, where X n x = G n x ∕ | G n x |. As applications we improve previous results on the large deviation principle for the matrix norm ‖ G n ‖ and obtain a precise local limit theorem with large deviations. [ABSTRACT FROM AUTHOR]

We study decreasing rearrangements of functions defined on (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by K > 0 and dimension bounded above by N ∈ (1 , ∞) in a synthetic sense, the so called CD (K , N) spaces. We first establish a Polya-Szego type inequality stating that the W 1 , p -Sobolev norm decreases under such a rearrangement and apply the result to show sharp spectral gap for the p -Laplace operator with Dirichlet boundary conditions (on open subsets), for every p ∈ (1 , ∞). This extends to the non-smooth setting a classical result of Bérard-Meyer [14] and Matei [41] ; remarkable examples of spaces fitting our framework and for which the results seem new include: measured-Gromov Hausdorff limits of Riemannian manifolds with Ricci ≥ K > 0 , finite dimensional Alexandrov spaces with curvature ≥ K > 0 , Finsler manifolds with Ricci ≥ K > 0. In the second part of the paper we prove new rigidity and almost rigidity results attached to the aforementioned inequalities, in the framework of RCD (K , N) spaces, which are interesting even for smooth Riemannian manifolds with Ricci ≥ K > 0. [ABSTRACT FROM AUTHOR]

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. We show that if the transition operator of the chain is operator uniformly integrable and the chain is ρ -mixing, then the limit is the same as if the summands were independent. We provide an example of a Markov chain that is operator uniformly integrable (and admits a spectral gap) while it is not hyperbounded. What makes our assumptions working is a new, efficient version of the Principle of Conditioning. [ABSTRACT FROM AUTHOR]

We study two types of Metropolis–Hastings (MH) reversiblizations for non-reversible Markov chains with Markov kernel P. While the first type is the classical Metropolised version of P , we introduce a new self-adjoint kernel which captures the opposite transition effect of the first type, that we call the second MH kernel. We investigate the spectral relationship between P and the two MH kernels. Along the way, we state a version of Weyl's inequality for the spectral gap of P (and hence its additive reversiblization), as well as an expansion of P. Both results are expressed in terms of the spectrum of the two MH kernels. In the spirit of Fill (1991) and Paulin (2015), we define a new pseudo-spectral gap based on the two MH kernels, and show that the total variation distance from stationarity can be bounded by this gap. We give variance bounds of the Markov chain in terms of the proposed gap, and offer spectral bounds in metastability and Cheeger's inequality in terms of the two MH kernels by comparison of Dirichlet form and Peskun ordering. [ABSTRACT FROM AUTHOR]

The Laplacian of a (weighted) Cayley graph on the Weyl group W (B n) is a N × N matrix with N = 2 n n ! equal to the order of the group. We show that for a class of (weighted) generating sets, its spectral gap (lowest nontrivial eigenvalue), is actually equal to the spectral gap of a 2 n × 2 n matrix associated to a 2 n -dimensional permutation representation of W n. This result can be viewed as an extension to W (B n) of an analogous result valid for the symmetric group, known as "Aldous' spectral gap conjecture", proven in 2010 by Caputo, Liggett and Richthammer. [ABSTRACT FROM AUTHOR]

Abstract We show that the tensor product M ⊗ ‾ N of any two full factors M and N (possibly of type III) is full and we compute Connes' invariant τ (M ⊗ ‾ N) in terms of τ (M) and τ (N). The key novelty is an enhanced spectral gap property for full factors of type III. Moreover, for full factors of type III with almost periodic states, we prove an optimal spectral gap property. As an application of our main result, we also show that for any full factor M and any non-type I amenable factor P , the tensor product factor M ⊗ ‾ P has a unique McDuff decomposition, up to stable unitary conjugacy. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the asymptotic behavior of solutions to the linear spatially homogeneous Boltzmann equation for hard potentials without angular cutoff. We obtain an optimal rate of exponential convergence towards equilibrium in a L 1-space with a polynomial weight. Our strategy is taking advantage of a spectral gap estimate in the Hilbert space L 2 (μ − 1 2 ) and a quantitative spectral mapping theorem developed by Gualdani et al. (2017). [ABSTRACT FROM AUTHOR]

Climatic or geologic controls, such as tectonics or glacial drainage, might impose constraints on landscape self-organization resulting in spatial patterns of rivers and valleys which do not obey the typical self-similar relationships found in most landscapes. The goal of this study is to quantify how such geologic constraints express themselves on channel network topology, spatial heterogeneity of drainage patterns, and emergence of preferred scales of landscape dissection. We use as an example a basin located in the Upper Midwestern United States where successive glaciations over the past thousand years have led to a pronounced spatially anisotropic channel network structure which defeats most scaling laws of fluvial landscapes. This is contrasted with another river basin in the North-Central U.S. which has been organized under the absence of major geologic influences and follows a typical self-similar channel network organization. We show how the geologic constraints have imposed a competition for space which is captured in the slope–local drainage density probabilistic structure, in the failure of self-similarity in basin-wide river network topology, and in the length-area scaling relationship being not typical of fluvial landscapes. Via a two-dimensional wavelet analysis and synthesis, we demonstrate the occurrence of a gap in the power spectrum which corresponds to the presence of preferred scales of organization, and characterize them through multi-scale detrending. The developed methodologies can be useful in advancing our geomorphologic understanding of how external controls might manifest themselves in creating a landscape dissection that is outside the norm and how this dissection can be studied objectively for understanding cause and effect. [ABSTRACT FROM AUTHOR]

Many finite-state reversible Markov chains can be naturally decomposed into “projection” and “restriction” chains. In this paper we provide bounds on the total variation mixing times of the original chain in terms of the mixing properties of these related chains. This paper is in the tradition of existing bounds on Poincaré and log-Sobolev constants of Markov chains in terms of similar decompositions (Jerrum et al., 2004; Madras and Randall, 2002; Martin and Randall, 2006; Madras and Yuen, 2009). Our proofs are simple, relying largely on recent results relating hitting and mixing times of reversible Markov chains (Peres and Sousi, 2013; Oliveira, 2012). We describe situations in which our results give substantially better bounds than those obtained by applying existing decomposition results and provide examples for illustration. [ABSTRACT FROM AUTHOR]

The present paper is devoted to estimating effective resistances in large weighted undirected Cayley graph Cay ( S n , T n ) on symmetric group S n . We obtain asymptotically exact bounds for minimal resistances in Cay ( S n , T n ) . Maximal and average resistances in Cayley graphs on symmetric groups are given in terms of degree, girth and spectral gap. Sharp asymptotic bounds for maximal and average resistances are obtained for star and transposition Cayley graphs. Finally, we find improved estimates of effective resistances in bubble-sort Cayley graphs. [ABSTRACT FROM AUTHOR]

Following recent work by Kollár and Sarnak, we study gaps in the spectra of large connected cubic and quartic graphs with minimum spectral gap. We focus on two sequences of graphs, denoted Δ n and Γ n which are more 'symmetric' compared to the other graphs in these two families, respectively. We prove that (1 , 5 ] is a gap interval for Δ n , and [ (− 1 + 17) / 2 , 3 ] is a gap interval for Γ n. We conjecture that these two are indeed maximal gap intervals. As a by-product, we show that the eigenvalues of Δ n lying in the interval [ − 3 , − 5 ] (in particular, its minimum eigenvalue) converge to (1 − 33) / 2 and the eigenvalues of Γ n lying in the interval [ − 4 , − (1 + 17) / 2 ] (and in particular, its minimum eigenvalue) converge to 1 − 13 as n tends to infinity. The proofs of the above results heavily depend on the following property which can be of independent interest: with few exceptions, all the eigenvalues of connected cubic and quartic graphs with minimum spectral gap are simple. [ABSTRACT FROM AUTHOR]

We establish metastability in the sense of Lebowitz and Penrose under practical and simple hypotheses for Markov chains on a finite configuration space in some asymptotic regime. By comparing restricted ensembles and quasi-stationary measures, and introducing soft measures as an interpolation between the two, we prove an asymptotic exponential exit law and, on a generally different time scale, an asymptotic exponential transition law. By using potential-theoretic tools, and introducing “ ( κ , λ ) -capacities”, we give sharp estimates on relaxation time, as well as mean exit time and transition time. We also establish local thermalization on shorter time scales. [ABSTRACT FROM AUTHOR]

An algorithm is presented for calculating eigenvalues lying in the spectral gaps of the essential spectrum of Schrödinger operators. The method is applicable to the scalar and matrix Schrödinger equations. Shooting methods and Floquet theory (including a new monotonicity result for the associated Atkinson Θ -matrices) are employed to deal with the λ -dependent BVPs. The proposed method shows results with higher accuracy and lower cost comparing with those obtained using finite difference schemes combined with a contour integral λ -nonlinear eigenvalue algorithm or using the dissipative barrier scheme with domain truncations which lead to a λ -linear eigenvalue problem. [ABSTRACT FROM AUTHOR]

This paper deals with the long time behaviour of solutions to the spatially homogeneous Landau equation with hard potentials. We prove an exponential in time convergence towards the equilibrium with the optimal rate given by the spectral gap of the associated linearised operator. This result improves the polynomial in time convergence obtained by Desvillettes and Villani [5] . Our approach is based on new decay estimates for the semigroup generated by the linearised Landau operator in weighted (polynomial or stretched exponential) L p -spaces, using a method developed by Gualdani, Mischler and Mouhot [7] . [ABSTRACT FROM AUTHOR]

Abstract: We consider the self-similar fragmentation equation with a superquadratic fragmentation rate and provide a quantitative estimate of the spectral gap. [Copyright &y& Elsevier]

The FELiChEM infrared free-electron laser consists of two oscillators: the mid-infrared oscillator and the far-infrared oscillator, covering the spectral range of 2– 40 μ m and 20– 200 μ m , respectively. The mid-infrared oscillator finished the commissioning in 2020 and a deep spectral gap around the wavelength of 21 μ m existed. In this paper, combining theoretical analysis and numerical simulation, the research on this waveguide effect has been carried out and the simulation results highly agree with the measurement. Through using the same method, we study the waveguide effect in the far-infrared oscillator for different waveguide sizes. The results show that, with the waveguide of 30 mm × 10 mm initially used, a spectral gap will be found at 89 μ m and the FEL power at the wavelength longer than 100 μ m will be rather low. Based on these studies, a reasonable waveguide size of 30 mm × 16 mm has been reselected to make the tunable range of the far-infrared oscillator covering 20– 200 μ m. The recent measurement carried out in 2021 also agree with the prediction of the simulation results. [ABSTRACT FROM AUTHOR]

Abstract: This work provides a description of the critical threshold phenomenon in multi-dimensional restricted Euler–Poisson (REP) equations, introduced in [H. Liu, E. Tadmor. Spectral dynamics of the velocity gradient field in restricted fluid flows, Comm. Math. Phys. 228 (2002) 435–466]. For three-dimensional REP equations, we identified both upper thresholds for the finite-time blow up of solutions and subthresholds for the global existence of solutions, with the thresholds depending on the relative size of the eigenvalues of the initial velocity gradient matrix and the initial density. For the attractive forcing case, these one-sided threshold conditions of the initial configurations are optimal, and the corresponding results also hold for arbitrary dimensions ( ). [Copyright &y& Elsevier]

Abstract: We prove a critical point theorem, which is an infinite dimensional generalization of the classical saddle point theorem of P. H. Rabinowitz. As an application we obtain solution of asymptotically linear Schrödinger equations with periodic potential. [Copyright &y& Elsevier]

Abstract: In this paper, we study the discrete nonlinear equation. where ω belongs to a finite spectral gap of the operator L or it is a lower edge of a finite spectral gap. By using a variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou, we will study the existence of nontrivial solitons for this equation with g n is super linear. [Copyright &y& Elsevier]

Abstract: For convergence rates of a time homogeneous Markov process, sufficient conditions are given in terms of an exponential -coupling. This provides sufficient conditions for convergence rates and related spectral and functional properties (spectral gap and Poincaré inequality) in terms of appropriate combination of ‘local mixing’ and ‘recurrence’ conditions on the initial process, typical in the ergodic theory of Markov processes. The range of applications of the approach includes processes that are not time-reversible. In particular, sufficient conditions for the spectral gap property for the Lévy driven Ornstein–Uhlenbeck process are established. [Copyright &y& Elsevier]

Abstract: Spectral clustering techniques are heuristic algorithms aiming to find approximate solutions to difficult graph-cutting problems, usually NP-complete, which are useful to clustering. A fundamental working hypothesis of these techniques is that the optimal partition of K classes can be obtained from the first K eigenvectors of the graph normalized Laplacian matrix L N if the gap between the K-th and the K+1-th eigenvalue of L N is sufficiently large. If the gap is small a perturbation may swap the corresponding eigenvectors and the results can be very different from the optimal ones. In this paper we suggest a weaker working hypothesis: the optimal partition of K classes can be obtained from a K-dimensional subspace of the first eigenvectors, where M is a parameter chosen by the user. We show that the validity of this hypothesis can be confirmed by the gap size between the K-th and the M+1-th eigenvalue of L N . Finally we present and analyse a simple probabilistic algorithm that generalizes current spectral techniques in this extended framework. This algorithm gives results on real world graphs that are close to the state of the art by selecting correct K-dimensional subspaces of the linear span of the first M eigenvectors, robust to small changes of the eigenvalues. [Copyright &y& Elsevier]

Abstract: We consider zero-range processes with jump rate for . We obtain that for the local process confined to a cube in of width , the spectral gap is bounded below by positive multiple of , where and is the total number of particles in the cube. [Copyright &y& Elsevier]

Abstract: In this paper we study the Riesz transform on complete and connected Riemannian manifolds M with a certain spectral gap in the spectrum of the Laplacian. We show that on such manifolds the Riesz transform is bounded for all . This generalizes a result by Mandouvalos and Marias and extends a result by Auscher, Coulhon, Duong, and Hofmann to the case where zero is an isolated point of the spectrum of the Laplacian. [Copyright &y& Elsevier]

Abstract: We consider a diffusion process on , which upon hitting , is redistributed in according to a probability measure depending continuously on its exit point. We prove that the distribution of the process converges exponentially fast to its unique invariant distribution and characterize the exponent as the spectral gap for a differential operator that serves as the generator of the process on a suitable function space. [Copyright &y& Elsevier]

Abstract: The whole of the modern treatment of wind engineering depends on the 1957 work by Van der Hoven in which he identified the macrometeorological spectrum, the micrometeorological spectrum and the spectral gap between them centred, roughly, on a frequency of 1cycle/h. Since then, very little work seems to have been done on the macrometeorological spectrum. This paper reports the analysis of a long (up to 30 years) record of hourly mean wind speeds recorded at 17m above ground on an open site at Boscombe Down, Wilts, UK. The broadband macrometeorological spectrum was very similar to that reported by Van der Hoven. Additionally, periodic deterministic components were identified corresponding to an annual cycle, a diurnal cycle and harmonics up to the sixth of the diurnal cycle. A model for the diurnal cycle is suggested which would produce these harmonic effects. This model implies that the diurnal cycle is intermittent and may be associated with the anti-cyclonic portion of the wind climate. The implications of the results on the derivation of design wind speeds are briefly discussed. [Copyright &y& Elsevier]

Abstract: In this paper we prove new constructive coercivity estimates for the Boltzmann collision operator without cutoff, that is for long-range interactions. In particular we give a generalized sufficient condition for the existence of a spectral gap which involves both the growth behavior of the collision kernel at large relative velocities and its singular behavior at grazing and frontal collisions. It provides in particular existence of a spectral gap and estimates on it for interactions deriving from the hard potentials , , or the so-called moderately soft potentials , (without angular cutoff). In particular this paper recovers (by constructive means), improves and extends previous results of Pao [Y.P. Pao, Boltzmann collision operator with inverse-power intermolecular potentials. I, Comm. Pure Appl. Math. 27 (1974) 407–428; Y.P. Pao, Boltzmann collision operator with inverse-power intermolecular potentials. II, Comm. Pure Appl. Math. 27 (1974) 559–581]. We also obtain constructive coercivity estimates for the Landau collision operator for the optimal coercivity norm pointed out in [Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys. 231 (2002) 391–434] and we formulate a conjecture about a unified necessary and sufficient condition for the existence of a spectral gap for Boltzmann and Landau linearized collision operators. [Copyright &y& Elsevier]

Abstract: Structures vibrate with discrete natural frequencies, which divide the frequency spectrum into frequency-free ranges, or spectral gaps. We may need a structure to have a particular spectral gap, and if this range is found to contain frequencies then the structure must be changed. The extent of the changes depends on the number of natural frequencies that are contained in the required spectral gap – the rank of the bracing must be at least the number of frequencies to be removed. This paper first deals with bracing of this minimal rank, connecting to the least number of freedoms possible, and later generalises this to higher rank bracing, with connections to any number of freedoms. The question is thus If a required spectral gap contains n natural frequencies, can changes to the structure stiffness of rank r ⩾ n, connecting to c ⩾ r freedoms i 1, i 2,…, i c remove them? In all cases, a simple criterion is developed for answering this question, and if the answer is yes, all successful changes are identified as mappings from more fundamental sets. The cases are developed separately even though later cases imply earlier ones, for clarity of the argument. [Copyright &y& Elsevier]

Abstract: A coupling method is used to obtain the explicit upper and lower bounds for convergence rates in strong ergodicity for Markov processes. For one-dimensional diffusion processes and birth–death processes, these bounds are sharp in the sense that the upper one and the lower one only differ in a constant. [Copyright &y& Elsevier]

Abstract: In this paper we study quadratic transportation cost inequalities. To this end we introduce new families of inequalities (for quadratic transportation cost and for relative entropy) that are shown to be equivalent to the Poincaré inequality. This allows us to give some examples of measures satisfying but not the logarithmic-Sobolev inequality. [Copyright &y& Elsevier]

Abstract: If a Brownian motion is physically constrained to the interval by reflecting it at the endpoints, one obtains an ergodic process whose exponential rate of convergence to equilibrium is . On the other hand, if Brownian motion is conditioned to remain in up to time t, then in the limit as one obtains an ergodic process whose exponential rate of convergence to equilibrium is . A recent paper [Grigorescu and Kang, J. Theoret. Probab. 15 (2002) 817–844] considered a different kind of physical constraint—when the Brownian motion reaches an endpoint, it is catapulted to the point , where , and then continues until it again hits an endpoint at which time it is catapulted again to , etc. The resulting process—Brownian motion physically returned to the point —is ergodic and the exponential rate of convergence to equilibrium is independent of p and equals . In this paper we define a conditioning analog of the process physically returned to the point and study its rate of convergence to equilibrium. [Copyright &y& Elsevier]

Suppose that for a real function f on the real line, the support of its Fourier transform is disjoint from an interval (−a,a). We prove the conjecture of B. Logan that under these assumptions the lower density of the sequence of sign changes of f is at least a/π. This result assumes a growth condition: f should be integrable with respect to a non-quasianalytic weight. We show that this growth restriction is best possible (for sufficiently regular weights). For functions and distributions f of faster growth, Logan''s conjecture does not hold, and for this class we prove a weaker result estimating the number of sign changes from below in terms of an averaged lower density. [Copyright &y& Elsevier]

We consider a two-dimensional (2D) convection model augmented with the rotational Coriolis forcing, Ut+U·∇xU=2kU⊥, with a fixed 2k being the inverse Rossby number. We ask whether the action of dispersive rotational forcing alone, U⊥, prevents the generic finite-time breakdown of the free nonlinear convection. The answer provided in this work is a conditional yes. Namely, we show that the rotating Euler equations admit global smooth solutions for a subset of generic initial configurations. With other configurations, however, finite-time breakdown of solutions may and actually does occur. Thus, global regularity depends on whether the initial configuration crosses an intrinsic, O(1) critical threshold (CT), which is quantified in terms of the initial vorticity, ω0=∇×U0, and the initial spectral gap associated with the 2×2 initial velocity gradient, η0≔λ2(0)−λ1(0),λj(0)=λj(∇U0). Specifically, global regularity of the rotational Euler equation is ensured if and only if 4kω0(α)+η20(α)<4k2,∀α∈R2. We also prove that the velocity field remains smooth if and only if it is periodic. An equivalent Lagrangian formulation reconfirms the CT and shows a global periodicity of velocity field as well as the associated particle orbits. Moreover, we observe yet another remarkable periodic behavior exhibited by the gradient of the velocity field. The spectral dynamics of the Eulerian formulation [SIAM J. Math. Anal. 33 (2001) 930] reveals that the vorticity and the divergence of the flow evolve with their own path-dependent period. We conclude with a kinetic formulation of the rotating Euler equation. [Copyright &y& Elsevier]

We study a model of a nonlinear lattice with an on-site potential that induces a breather gap, i.e. a frequency interval where breathers are unstable. The gap is determined numerically and after thermalization of the system we compute power spectra of local displacements. We find that the gap is clearly manifested at small couplings through a large spectral contribution in the linearized phonon region that is not shifted strongly with temperature. [Copyright &y& Elsevier]

We prove a uniform Poincare´ inequality for non-interacting unbounded spin systems with a conservation law, when the single-site potential is a bounded perturbation of a convex function with polynomial growth at infinity. The result is then applied to Ginzburg–Landau processes to show diffusive scaling of the associated spectral gap. [Copyright &y& Elsevier]

In this paper we study the relationships between two Markov Chain Monte Carlo algorithms—the Swapping Algorithm (also known as the Metropolis-coupled algorithm) and the simulated tempering algorithm. We give a proof that the spectral gap of the simulated tempering chain is bounded below by a multiple of that of the swapping chain. [Copyright &y& Elsevier]