Poincaré and logarithmic Sobolev constants for metastable Markov chains via capacitary inequalities

Schlichting, André; Slowik, Martin

doi:10.1214/19-aap1484

Cited by 6 publications

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References 46 publications

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“…The representation ( 34 ) is fruitful to many more applications in which the components of the mixture do not need to be absolutely continuous. Similar ideas for estimating mean-differences were successfully applied in the metastable setting [ 2 , 14 ], in which suitable bounds on the

-norm are obtained. In this regard, the bound ( 34 ) promises many interesting new insights for future studies.…”

Section: Discussionmentioning

confidence: 99%

Poincaré and Log–Sobolev Inequalities for Mixtures

Schlichting

2019

Entropy

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This work studies mixtures of probability measures on R n and gives bounds on the Poincaré and the log-Sobolev constant of two-component mixtures provided that each component satisfies the functional inequality, and both components are close in the χ 2 -distance. The estimation of those constants for a mixture can be far more subtle than it is for its parts. Even mixing Gaussian measures may produce a measure with a Hamiltonian potential possessing multiple wells leading to metastability and large constants in Sobolev type inequalities. In particular, the Poincaré constant stays bounded in the mixture parameter whereas the log-Sobolev may blow up as the mixture ratio goes to 0 or 1. This observation generalizes the one by Chafaï and Malrieu to the multidimensional case. The behavior is shown for a class of examples to be not only a mere artifact of the method.

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-norm are obtained. In this regard, the bound ( 34 ) promises many interesting new insights for future studies.…”

Section: Discussionmentioning

confidence: 99%

Poincaré and Log–Sobolev Inequalities for Mixtures

Schlichting

2019

Entropy

Self Cite

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“…The advantage of this definition compared to the one given in Bovier and den Hollander [5,Definition 8.2] is twofold: it allows for direct control of ℓ p (µ N )norms of functions, and does not depend on the cardinality of the state space. For a more detailed comparison of the two definitions of metastability we refer to [19,Remark 1.2].…”

Section: Example 23 (Randomly Diluted Hopfield Model)mentioning

confidence: 99%

“…By using arguments similar to those given in [19,Lemma 4.1], it follows that {M 1,N , M 2,N } forms a pair of metastable sets in the sense of Definition 2.4.…”

Section: Example 23 (Randomly Diluted Hopfield Model)mentioning

confidence: 99%

“…, M K,N }, we can decompose the state space S N into the domains of attraction with to respect the dynamics of the annealed model. More precisely, by following [19,Definition 1.4], within the event Ω meta (N ) the metastable sets {M 1,N , . .…”

Section: Example 23 (Randomly Diluted Hopfield Model)mentioning

confidence: 99%

“…See, for instance, Bianchi, Bovier and Ioffe [2] and [8], where long and model-dependent computations were needed to prove that the relevant contribution of the harmonic sum is localised around the starting metastable set. Schlichting and Slowik in [19], using an alternative definition of metastable sets, prove that localisation of the harmonic sum around the starting metastable set holds in large generality. Their work allows us to derive results for a large class of models.…”

Section: Introductionmentioning

confidence: 99%

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Metastability for Glauber Dynamics on the Complete Graph with Coupling Disorder

Bovier

Hollander

Marello

2022

Commun. Math. Phys.

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Consider the complete graph on n vertices. To each vertex assign an Ising spin that can take the values $$-1$$ - 1 or $$+1$$ + 1 . Each spin $$i \in [n]=\{1,2,\dots , n\}$$ i ∈ [ n ] = { 1 , 2 , ⋯ , n } interacts with a magnetic field $$h \in [0,\infty )$$ h ∈ [ 0 , ∞ ) , while each pair of spins $$i,j \in [n]$$ i , j ∈ [ n ] interact with each other at coupling strength $$n^{-1} J(i)J(j)$$ n - 1 J ( i ) J ( j ) , where $$J=(J(i))_{i \in [n]}$$ J = ( J ( i ) ) i ∈ [ n ] are i.i.d. non-negative random variables drawn from a probability distribution with finite support. Spins flip according to a Metropolis dynamics at inverse temperature $$\beta \in (0,\infty )$$ β ∈ ( 0 , ∞ ) . We show that there are critical thresholds $$\beta _c$$ β c and $$h_c(\beta )$$ h c ( β ) such that, in the limit as $$n\rightarrow \infty $$ n → ∞ , the system exhibits metastable behaviour if and only if $$\beta \in (\beta _c, \infty )$$ β ∈ ( β c , ∞ ) and $$h \in [0,h_c(\beta ))$$ h ∈ [ 0 , h c ( β ) ) . Our main result is a sharp asymptotics, up to a multiplicative error $$1+o_n(1)$$ 1 + o n ( 1 ) , of the average crossover time from any metastable state to the set of states with lower free energy. We use standard techniques of the potential-theoretic approach to metastability. The leading order term in the asymptotics does not depend on the realisation of J, while the correction terms do. The leading order of the correction term is $$\sqrt{n}$$ n times a centred Gaussian random variable with a complicated variance depending on $$\beta ,h$$ β , h , on the law of J and on the metastable state. The critical thresholds $$\beta _c$$ β c and $$h_c(\beta )$$ h c ( β ) depend on the law of J, and so does the number of metastable states. We derive an explicit formula for $$\beta _c$$ β c and identify some properties of $$\beta \mapsto h_c(\beta )$$ β ↦ h c ( β ) . Interestingly, the latter is not necessarily monotone, meaning that the metastable crossover may be re-entrant.

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Spectral Analysis of Discrete Metastable Diffusions

Gesù

2023

Commun. Math. Phys.

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We consider a discrete Schrödinger operator $$ H_\varepsilon = -\varepsilon ^2\Delta _\varepsilon + V_\varepsilon $$ H ε = - ε 2 Δ ε + V ε on $$\ell ^2(\varepsilon \mathbb {Z}^d)$$ ℓ 2 ( ε Z d ) , where $$\varepsilon >0$$ ε > 0 is a small parameter and the potential $$V_\varepsilon $$ V ε is defined in terms of a multiwell energy landscape f on $$\mathbb {R}^d$$ R d . This operator can be seen as a discrete analog of the semiclassical Witten Laplacian of $$\mathbb {R}^d$$ R d . It is unitarily equivalent to the generator of a diffusion on $$\varepsilon \mathbb {Z}^d$$ ε Z d , satisfying the detailed balance condition with respect to the Boltzmann weight $$\exp {(-f/\varepsilon )}$$ exp ( - f / ε ) . These type of diffusions exhibit metastable behavior and arise in the context of disordered mean field models in Statistical Mechanics. We analyze the bottom of the spectrum of $$H_\varepsilon $$ H ε in the semiclassical regime $$\varepsilon \ll 1$$ ε ≪ 1 and show that there is a one-to-one correspondence between exponentially small eigenvalues and local minima of f. Then we analyze in more detail the bistable case and compute the precise asymptotic splitting between the two exponentially small eigenvalues. Through this purely spectral-theoretical analysis of the discrete Witten Laplacian we recover in a self-contained way the Eyring–Kramers formula for the metastable tunneling time of the underlying stochastic process.

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Poincaré and logarithmic Sobolev constants for metastable Markov chains via capacitary inequalities

Cited by 6 publications

References 46 publications

Poincaré and Log–Sobolev Inequalities for Mixtures

Poincaré and Log–Sobolev Inequalities for Mixtures

Metastability for Glauber Dynamics on the Complete Graph with Coupling Disorder

Spectral Analysis of Discrete Metastable Diffusions

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