Modified log-Sobolev inequalities, Beckner inequalities and moment estimates

Adamczak, Radosław; Polaczyk, Bartłomiej; Strzelecki, Michał

doi:10.1016/j.jfa.2021.109349

Cited by 8 publications

(16 citation statements)

References 106 publications

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“…where f [1] p (•, •) is the double sum operator associated with the divided difference of the function:…”

Section: Preliminariesmentioning

confidence: 99%

“…In Section 3.3, we first show in Proposition 3.19 that the family of p-Beckner's inequalities may imply a better mixing time bound than the ones in [62,89]. In Proposition 3.21, we extend [1,Proposition 3.3] for the classical case to the quantum setting and obtain moment estimates under (Bec-p). Then we analyze the quantum Beckner constant for the depolarizing semigroup and reduce it to the computation of the classical one for a Markov chain on the two-point space; see Proposition 3.17.…”

Section: Introductionmentioning

confidence: 99%

“…For instance, Dai Pra et al [37] provided an example where the MLSI (1.3) holds while the LSI (1.6) fails. In what follows, to avoid confusion between (1.5) and (1.7), following [1] and [35] we call the inequality of the form (1.7) the dual Beckner's inequality. Similarly to the diffusion case, the Bakry-Émery method and gradient flow techniques are two main approaches for the validity of (1.2).…”

Section: Introductionmentioning

confidence: 99%

“…In this work we try to close this gap. We consider the primitive QMS with σ-detailed balance condition for σ ∈ D + (H), and define the family of quantum p-Beckner's inequalities and their dual version, by extending the definitions in [1,35,64] for classical Markov semigroups; see Definition 3.1 for the functional inequalities that we will mainly focus on. It turns out that the p-Beckner's inequality (Bec-p) describes the rate at which the quantum p-divergence F p,σ (2.30), which can be viewed as the normalized noncommutative L p -norm, tends to zero.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Interpolation between modified logarithmic Sobolev and Poincare inequalities for quantum Markovian dynamics

Li¹,

Lu²

2022

Preprint

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In this work, we define the quantum p-divergences and introduce the Beckner's inequalities for the primitive quantum Markov semigroups with σ-detailed balance condition. Such inequalities quantify the convergence rate of the quantum dynamics in the noncommutative Lp-norm. We obtain a number of implications between Beckner's inequalities and other quantum functional inequalities. In particular, we show that the quantum Beckner's inequalities interpolate between the Sobolev-type inequalities and the Poincaré inequality in a sharp way. We provide a uniform lower bound for the Beckner constants αp in terms of the spectral gap, and establish the stability of αp with respect to the invariant state. As applications of Beckner's inequalities, we obtain an improved bound for the mixing time. For the symmetric quantum Markov semigroups, we derive moment estimates, which further implies a concentration inequality.We introduce a new class of quantum transport distances W 2,p interpolating the quantum 2-Wasserstein distance by Carlen and Maas [J. Funct. Anal. 273(5), 1810-1869] and a noncommutative Ḣ−1 Sobolev distance. We show that the quantum Markov semigroup with σ-detailed balance is the gradient flow of the quantum p-divergence with respect to the metric W 2,p . We prove that the set of quantum states equipped with W 2,p is a complete geodesic space. We then consider the associated entropic Ricci curvature lower bound via the geodesic convexity of p-divergence, and obtain an HWI-type interpolation inequality. This enables us to prove that the positive Ricci curvature implies the quantum Beckner's inequality, from which a transport cost and the Poincaré inequalities can follow.

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“…where f [1] p (•, •) is the double sum operator associated with the divided difference of the function:…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Interpolation between modified logarithmic Sobolev and Poincare inequalities for quantum Markovian dynamics

Li¹,

Lu²

2022

Preprint

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show abstract

“…Furthermore, a special class of non-Lipschitz functions for which concentration bounds based on inequalities of type (5.1) and (5.3) have been established in recent years are so-called multilinear polynomials, i. e. polynomials which are affine with respect to each variable. See for instance [19,Theorem 5] and [1,Corollary 5.4]. Finally, note that (5.2) is frequently used in the context of Markov processes, and it can be shown to be equivalent to exponential decay of the relative entropy along the Glauber semigroup (cf.…”

Section: Logarithmic Sobolev Inequalitiesmentioning

confidence: 99%

Large deviations and a phase transition in the Block Spin Potts models

Knöpfel,

Löwe,

Sambale

2020

Preprint

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We introduce and analyze a generalization of the blocks spin Ising (Curie-Weiss) models that were discussed in a number of recent articles. In these block spin models each spin in one of s blocks can take one of a finite number of q ≥ 3 values, hence the name block spin Potts model. The values a spin can take are called colors. We prove a large deviation principle for the percentage of spins of a certain color in a certain block. These values are represented in an s × q matrix. We show that for uniform block sizes and appropriately chosen interaction strength there is a phase transition. In some regime the only equilibrium is the uniform distribution of all colors in all blocks, while in other parameter regimes there is one predominant color, and this is the same color with the same frequency for all blocks. Finally, we establish log-Sobolev-type inequalities for the block spin Potts model.

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Phase transitions and noise sensitivity on the Poisson space via stopping sets and decision trees

Last

Peccati

Yogeshwaran

2023

Random Struct Algorithms

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Proofs of sharp phase transition and noise sensitivity in percolation have been significantly simplified by the use of randomized algorithms, via the OSSS inequality (proved by O'Donnell et al. and the Schramm–Steif inequality for the Fourier‐Walsh coefficients of functions defined on the Boolean hypercube. In this article, we prove intrinsic versions of the OSSS and Schramm–Steif inequalities for functionals of a general Poisson process, and apply these new estimates to deduce sufficient conditions—expressed in terms of randomized stopping sets—yielding sharp phase transitions, quantitative noise sensitivity, exceptional times and bounds on critical windows for monotonic Boolean Poisson functions. Our analysis is based on a new general definition of “stopping set”, not requiring any topological property for the underlying measurable space, as well as on the new concept of a “continuous‐time decision tree”, for which we establish several fundamental properties. We apply our findings to the k$$ k $$‐percolation of the Poisson Boolean model and to the Poisson‐based confetti percolation with bounded random grains. In these two models, we reduce the proof of sharp phase transitions for percolation, and of noise sensitivity for crossing events, to the construction of suitable randomized stopping sets and the computation of one‐arm probabilities at criticality. This enables us to settle an open problem suggested by Ahlberg et al. (a special case of which was conjectured earlier by Ahlberg et al. on noise sensitivity of crossing events for the planar Poisson Boolean model with random balls whose radius distribution has finite false(2+αfalse)$$ \left(2+\alpha \right) $$‐moments and also show the same for planar confetti percolation model with bounded random balls. We also prove that critical probability is 1false/2$$ 1/2 $$ for the planar confetti percolation model with bounded, πfalse/2$$ \pi /2 $$‐rotation invariant and reflection invariant random grains. Such a result was conjectured by Benjamini and Schramm in the case of fixed balls and proved by Müller, Hirsch and Ghosh and Roy in the case of balls, boxes and random boxes, respectively; our results contain all previous findings as special cases.

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Modified log-Sobolev inequalities, Beckner inequalities and moment estimates

Cited by 8 publications

References 106 publications

Interpolation between modified logarithmic Sobolev and Poincare inequalities for quantum Markovian dynamics

Interpolation between modified logarithmic Sobolev and Poincare inequalities for quantum Markovian dynamics

Large deviations and a phase transition in the Block Spin Potts models

Phase transitions and noise sensitivity on the Poisson space via stopping sets and decision trees

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