Spectral Gap Estimates Research Articles

Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff

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 ϕ ( r ) = r − ( s − 1 ) , s ⩾ 5 , or the so-called moderately soft potentials ϕ ( r ) = r − ( s − 1 ) , 3 < s < 5 (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.

Small-world MCMC and convergence to multi-modal distributions: From slow mixing to fast mixing

We compare convergence rates of Metropolis–Hastings chains to multi-modal target distributions when the proposal distributions can be of “local” and “small world” type. In particular, we show that by adding occasional long-range jumps to a given local proposal distribution, one can turn a chain that is “slowly mixing” (in the complexity of the problem) into a chain that is “rapidly mixing.” To do this, we obtain spectral gap estimates via a new state decomposition theorem and apply an isoperimetric inequality for log-concave probability measures. We discuss potential applicability of our result to Metropolis-coupled Markov chain Monte Carlo schemes.

Explicit Coercivity Estimates for the Linearized Boltzmann and Landau Operators

We prove explicit coercivity estimates for the linearized Boltzmann and Landau operators, for a general class of interactions including any inverse-power law interactions, and hard spheres. The functional spaces of these coercivity estimates depend on the collision kernel of these operators. They cover the spectral gap estimates for the linearized Boltzmann operator with Maxwell molecules, improve these estimates for hard potentials, and are the first explicit coercivity estimates for soft potentials (including in particular the case of Coulombian interactions). We also prove a regularity property for the linearized Boltzmann operator with non locally integrable collision kernels, and we deduce from it a new proof of the compactness of its resolvent for hard potentials without angular cutoff.

Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials

This paper deals with explicit spectral gap estimates for the linearized Boltzmann operator with hard potentials (and hard spheres). We prove that it can be reduced to the Maxwellian case, for which explicit estimates are already known. Such a method is constructive, does not rely on Weyl's Theorem and thus does not require Grad's splitting. The more physical idea of the proof is to use geometrical properties of the whole collision operator. In a second part, we use the fact that the Landau operator can be expressed as the limit of the Boltzmann operator as collisions become grazing in order to deduce explicit spectral gap estimates for the linearized Landau operator with hard potentials.

Spectral gap estimates for interacting particle systems via a Bochner-type identity

We develop a general technique, based on a Bochner-type identity, to estimate spectral gaps of a class of Markov operator. We apply this technique to various interacting particle systems. In particular, we give a simple and short proof of the diffusive scaling of the spectral gap of the Kawasaki model at high temperature. Similar results are derived for Kawasaki-type dynamics in the lattice without exclusion, and in the continuum. New estimates for Glauber-type dynamics are also obtained.

Estimate of spectral gap for continuous gas

Consider the continuous gas in a bounded domain Λ of Rd, described by a Gibbsian measure μΛη associated with a pair interaction ϕ, the inverse temperature β, the activity z>0, and the boundary condition η. When ϕ is nonnegative, we show that the spectral gap of a Glauber type dynamic (i.e., some Markov process reversible with respect to μΛη) in L2(μΛη) is bounded from below by 1−z∫Rd|1−e−βϕ(y)|dy and from above by 1+z∫Rd|1−e−βϕ(y)|dy, independent of Λ and η. This result improves a previous work by L. Bertini et al. (2002) and is extended also to the hard core case. Our approach consists to approximate the continuous gas model by the discrete spin model and to apply the M-ε theorem of Ligget. Some other results such as uniqueness, exponential convergence of the Glauber dynamic w.r.t. norms of Ligget's type are also obtained. On considère un gaz continu dans un domaine borné Λ de Rd, décrit par la mesure de Gibbs μΛη associée à l'interaction paire ϕ, la température inverse β, l'activité z>0 et la condition au bord η. Quand ϕ⩾0, nous démontrons que le trou spectral λ1 d'une dynamique du type Glauber (i.e., un processus de Markov réversible par rapport à μΛη) dans L2(μΛη) vérifie 1−z∫Rd|1−e−βϕ(y)|dy⩽λ1⩽1+z∫Rd|1−e−βϕ(y)|dy, indépendamment de Λ et de η. Ce résultat améliore le travail de Bertini et al. (2002), et est généralisé au cas de corps durs. Notre méthode consiste à approcher le gaz continu par un modèle de spin discret auquel le théorème M-ε de Ligget s'applique. Nous établissons également l'unicité de la dynamique de Glauber et sa convergence exponentielle par rapport aux normes du type Ligget.

Estimate of spectral gap for continuous gas

Consider the continuous gas in a bounded domain Λ of R d , described by a Gibbsian measure μ Λ η associated with a pair interaction ϕ, the inverse temperature β, the activity z>0, and the boundary condition η. When ϕ is nonnegative, we show that the spectral gap of a Glauber type dynamic (i.e., some Markov process reversible with respect to μ Λ η ) in L 2( μ Λ η ) is bounded from below by 1−z∫ R d |1−e −βϕ(y)| dy and from above by 1+z∫ R d |1−e −βϕ(y)| dy , independent of Λ and η. This result improves a previous work by L. Bertini et al. (2002) and is extended also to the hard core case. Our approach consists to approximate the continuous gas model by the discrete spin model and to apply the M- ε theorem of Ligget. Some other results such as uniqueness, exponential convergence of the Glauber dynamic w.r.t. norms of Ligget's type are also obtained.

Right order spectral gap estimates for generating sets of ℤ4

Using coupling arguments, a distance method and Zeifman's method we give sharp estimates on the spectral gap for a special case of the class of Markov chains on generating n-tuples of Abelian groups. In our case the group is ℤ4. © 2002 Wiley Periodicals, Inc. Random Struct. Alg. 20: 220–238, 2002

Right order spectral gap estimates for generating sets of ℤ4

AbstractUsing coupling arguments, a distance method and Zeifman's method we give sharp estimates on the spectral gap for a special case of the class of Markov chains on generating n‐tuples of Abelian groups. In our case the group is ℤ4. © 2002 Wiley Periodicals, Inc. Random Struct. Alg. 20: 220–238, 2002

On spectral gap estimates of a Markov chain via hitting times and coupling

Well-known inequalities for the spectral gap of a discrete-time Markov chain, such as Poincaré's and Cheeger's inequalities, do not perform well if the transition graph of the Markov chain is strongly connected. For example in the case of nearest-neighbour random walk on then-dimensional cube Poincaré's and Cheeger's inequalities are off by a factorn. Using a coupling technique and a contraction principle lower bounds on the spectral gap can be derived. Finally, we show that using the contraction principle yields a sharp estimate for nearest-neighbour random walk on then-dimensional cube.

On spectral gap estimates of a Markov chain via hitting times and coupling

Well-known inequalities for the spectral gap of a discrete-time Markov chain, such as Poincaré's and Cheeger's inequalities, do not perform well if the transition graph of the Markov chain is strongly connected. For example in the case of nearest-neighbour random walk on then-dimensional cube Poincaré's and Cheeger's inequalities are off by a factorn. Using a coupling technique and a contraction principle lower bounds on the spectral gap can be derived. Finally, we show that using the contraction principle yields a sharp estimate for nearest-neighbour random walk on then-dimensional cube.

On spectral gap estimates of a Markov chain via hitting times and coupling

Well-known inequalities for the spectral gap of a discrete-time Markov chain, such as Poincaré's and Cheeger's inequalities, do not perform well if the transition graph of the Markov chain is strongly connected. For example in the case of nearest-neighbour random walk on the n-dimensional cube Poincaré's and Cheeger's inequalities are off by a factor n. Using a coupling technique and a contraction principle lower bounds on the spectral gap can be derived. Finally, we show that using the contraction principle yields a sharp estimate for nearest-neighbour random walk on the n-dimensional cube.

Spectral gap estimates on compact manifolds

For a compact Riemannian manifold with boundary, its mass gap is the difference between the first and second smallest Dirichlet eigenvalues. In this paper, taking a variational approach, we obtain an explicit lower bound estimate of the mass gap for any compact manifold in terms of geometric quantities.

Non-Gaussian surface pinned by a weak potential

We consider a model of a two-dimensional interface of the SOS type, with finite-range, even, strictly convex, twice continuously differentiable interactions. We prove that, under an arbitrarily weak potential favouring zero-height, the surface has finite mean square heights. We consider the cases of both square well and $\delta$ potentials. These results extend previous results for the case of nearest-neighbours Gaussian interactions in \cite{DMRR} and \cite{BB}. We also obtain estimates on the tail of the height distribution implying, for example, existence of exponential moments. In the case of the $\delta$ potential, we prove a spectral gap estimate for linear functionals. We finally prove exponential decay of the two-point function (1) for strong $\delta$-pinning and the above interactions, and (2) for arbitrarily weak $\delta$-pinning, but with finite-range Gaussian interactions.

Estimation of spectral gap for elliptic operators

A variational formula for the lower bound of the spectral gap of an elliptic operator is presented in the paper for the first time. The main known results are either recovered or improved. A large number of new examples with sharp estimate are illustrated. Moreover, as an application of the march coupling, the Poincaré inequality with respect to the absolute distribution of the process is also studied.

Geometric Approaches to the Estimation of the Spectral Gap of Reversible Markov Chains

In this paper we consider the problem of estimating the spectral gap of a reversible Markov chain in terms of geometric quantities associated with the underlying graph. This quantity provides a bound on the rate of convergence of a Markov chain towards its stationary distribution. We give a critical and systematic treatment of this subject, summarizing and comparing the results of the two main approaches in the literature, algebraic and functional. The usefulness and drawbacks of these bounds are also discussed here.