## Spectral gap estimates for interacting particle systems via a Bochner type inequality (2006)

Venue: | J. Funct. Anal |

Citations: | 4 - 0 self |

### BibTeX

@ARTICLE{Boudou06spectralgap,

author = {Anne-severine Boudou and Pietro Caputo and Paolo Dai Pra and Gustavo Posta},

title = {Spectral gap estimates for interacting particle systems via a Bochner type inequality},

journal = {J. Funct. Anal},

year = {2006},

pages = {222--258}

}

### OpenURL

### Abstract

We develop a general technique, based on the Bakry–Emery approach, 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. 1 1

### Citations

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Citation Context ...equalities for interacting particle systems ([11]) has been motivated by both theoretical and computational purposes, and has led to the development of a rather sophisticated mathematical technology (=-=[15, 16, 12, 13, 4]-=-). The main aim of this paper is to adapt to a class of Markov processes with discontinuous trajectories, including many interesting interacting particle systems, an approach to functional inequalitie... |

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Citation Context ...dapt to a class of Markov processes with discontinuous trajectories, including many interesting interacting particle systems, an approach to functional inequalities that goes back to Bakry & Emery in =-=[1]-=- and that has been widely exploited for diffusion operators [7, 8, 10]. In the case of Poincaré inequality, this approach leads to the following proposition. Proposition 1.1 Under the irreducibility a... |

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Citation Context ...equalities for interacting particle systems ([11]) has been motivated by both theoretical and computational purposes, and has led to the development of a rather sophisticated mathematical technology (=-=[15, 16, 12, 13, 4]-=-). The main aim of this paper is to adapt to a class of Markov processes with discontinuous trajectories, including many interesting interacting particle systems, an approach to functional inequalitie... |

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Citation Context ...equalities for interacting particle systems ([11]) has been motivated by both theoretical and computational purposes, and has led to the development of a rather sophisticated mathematical technology (=-=[15, 16, 12, 13, 4]-=-). The main aim of this paper is to adapt to a class of Markov processes with discontinuous trajectories, including many interesting interacting particle systems, an approach to functional inequalitie... |

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Citation Context ...ined in [6], in the case of uniformly bounded interaction. The method in this paper allows unbounded interaction too. For the models in Section 7, estimates on the spectral gap were obtained first in =-=[2]-=-, via an inductive argument, and then in [9] via the same Bakry-Emery-type computation we use here; our point here is to show that this argument is a special case of a general, and rather powerful, me... |

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Citation Context ...usive scaling of the spectral gap of the Kawasaki model at sufficiently high temperature. This result goes back to Lu and Yau in [12]; their extremely difficult proof has been made more accessible in =-=[3]-=-, even though it still required a long and technical inductive argument. The statement proved in [12] and [3] is that diffusive scaling of the spectral gap follows from the so-called strong mixing con... |

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Citation Context ...ries, including many interesting interacting particle systems, an approach to functional inequalities that goes back to Bakry & Emery in [1] and that has been widely exploited for diffusion operators =-=[7, 8, 10]-=-. In the case of Poincaré inequality, this approach leads to the following proposition. Proposition 1.1 Under the irreducibility assumption E(f, f) = 0 ⇐⇒ ν[f; f] = 0, (1.4) the inequality is equivale... |

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Citation Context ... x∼z e −1 2 β∇xzH(η) ∇xzf(η), where the sum ∑ x∼z ranges over pairs x, z ∈ Λ with |x −z| = 1. In the case the potential Φ is of finite range, i.e. ΦA ≡ 0 up to a finite number of sets A, Lemma 4.3 in =-=[17]-=- can be used in a standard way to connect the gap of Ln.n. with that of L, getting the following result. Corollary 3.3 Let diam(Λ) = max{|x − z| : x, z ∈ Λ}, and assume Φ is a finite range potential. ... |

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Citation Context ..., then (1.3) holds with α � s/4; if (1.3) holds with α > 0, then (1.1) holds with k � α/2. Various consequences of (1.2) and (1.3) in terms of ergodicity of the semigroup Tt can be obtained (see e.g. =-=[6]-=-). For instance, under some additional conditions on the domain D(L), the modified logarithmic-Sobolev inequality is equivalent to the statement Entν(Ttf) � e −αt Entν(f) for each f with finite entrop... |

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Citation Context ...d interaction. The method in this paper allows unbounded interaction too. For the models in Section 7, estimates on the spectral gap were obtained first in [2], via an inductive argument, and then in =-=[9]-=- via the same Bakry-Emery-type computation we use here; our point here is to show that this argument is a special case of a general, and rather powerful, method. 2 General scheme In this section we gi... |

13 |
Hypercontractivity and spectral gap of symmetric diffusions with applications to the stochastic Ising models
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Citation Context ...ries, including many interesting interacting particle systems, an approach to functional inequalities that goes back to Bakry & Emery in [1] and that has been widely exploited for diffusion operators =-=[7, 8, 10]-=-. In the case of Poincaré inequality, this approach leads to the following proposition. Proposition 1.1 Under the irreducibility assumption E(f, f) = 0 ⇐⇒ ν[f; f] = 0, (1.4) the inequality is equivale... |

9 | The logarithmic Sobolev constant of Kawasaki dynamics under a mixing condition revisited
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Citation Context |

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Citation Context ...∑ ‖|Φ‖| ν [ c(η, zu) (∇zuf(η)) 2] , from which (3.6) follows. u,z u,z 4 Random walks on the complete graph Random walks on the complete graph interacting via a zero–range potential were considered in =-=[5]-=-. It was shown that the spectral gap of the process is positive as soon as a uniform log– concavity assumption is satisfied. Here we consider the case where we add a non–zero–range interaction to the ... |

9 |
Interacting particle systems. 276
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Citation Context ...rsion coincide, while for Markov processes with discontinuous trajectories the two inequalities are, in general, non equivalent. The study of functional inequalities for interacting particle systems (=-=[11]-=-) has been motivated by both theoretical and computational purposes, and has led to the development of a rather sophisticated mathematical technology ([15, 16, 12, 13, 4]). The main aim of this paper ... |

8 | Spectral gap for the zero range process with constant rate. preprint, Math Arkiv PR/0405161
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(Show Context)
Citation Context ...udes the interesting case of constant rates where gx(k) = 1 for all k � 1. (4.16) It is worthwhile observing that in this case if H = 0 the gap is of order (1+ρ) −2 with ρ = N/n, as recently shown in =-=[14]-=- by Morris. Note that the choice (4.16) makes the reference measure ¯ν N Vn uniform over SN. Thus (4.15) proves that the addition of a small mass (a > 0) is sufficient to give a density–independent lo... |

3 |
Remarks on Decay of Correlations and
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(Show Context)
Citation Context ...ries, including many interesting interacting particle systems, an approach to functional inequalities that goes back to Bakry & Emery in [1] and that has been widely exploited for diffusion operators =-=[7, 8, 10]-=-. In the case of Poincaré inequality, this approach leads to the following proposition. Proposition 1.1 Under the irreducibility assumption E(f, f) = 0 ⇐⇒ ν[f; f] = 0, (1.4) the inequality is equivale... |