Evolution of Concentration Under Lattice Spin-Flip Dynamics

AbstractWe consider spin-flip dynamics of Ising lattice spin systems and study the time evolution of concentration inequalities. For “weakly interacting” dynamics we show that the Gaussian concentration bound is conserved in the course of time and it is satisfied by the unique stationary Gibbs measure. Next we show that, for a general class of translation-invariant spin-flip dynamics, it is impossible to evolve in finite time from a low-temperature Gibbs state towards a measure satisfying the Gaussian concentration bound. Finally, we consider the time evolution of the weaker uniform variance bound, and show that this bound is conserved under a general class of spin-flip dynamics.

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Effective field renormalization group approach for Ising lattice spin systems

Journal of Magnetism and Magnetic Materials ◽

10.1016/0304-8853(94)90008-6 ◽

1994 ◽

Vol 131 (1-2) ◽

pp. 43-53 ◽

Cited By ~ 24

Author(s):

Ivon P. Fittipaldi

Keyword(s):

Renormalization Group ◽

Effective Field ◽

Spin Systems ◽

Ising Lattice ◽

Group Approach ◽

Lattice Spin ◽

Lattice Spin Systems ◽

Renormalization Group Approach ◽

Field Renormalization

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Correlation asymptotics for non-translation invariant lattice spin systems

Mathematische Nachrichten ◽

10.1002/mana.200610638 ◽

2008 ◽

Vol 281 (5) ◽

pp. 721-759 ◽

Cited By ~ 5

Author(s):

Oliver Matte

Keyword(s):

Spin Systems ◽

Translation Invariant ◽

Lattice Spin ◽

Lattice Spin Systems

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Approximate Tensorization of the Relative Entropy for Noncommuting Conditional Expectations

Annales Henri Poincaré ◽

10.1007/s00023-021-01088-3 ◽

2021 ◽

Author(s):

Ivan Bardet ◽

Ángela Capel ◽

Cambyse Rouzé

Keyword(s):

Relative Entropy ◽

Von Neumann Algebras ◽

Logarithmic Sobolev Inequality ◽

Spin Systems ◽

Von Neumann ◽

Strong Subadditivity ◽

Finite Dimensional ◽

Lattice Spin ◽

Lattice Spin Systems ◽

Conditional Expectations

AbstractIn this paper, we derive a new generalisation of the strong subadditivity of the entropy to the setting of general conditional expectations onto arbitrary finite-dimensional von Neumann algebras. This generalisation, referred to as approximate tensorization of the relative entropy, consists in a lower bound for the sum of relative entropies between a given density and its respective projections onto two intersecting von Neumann algebras in terms of the relative entropy between the same density and its projection onto an algebra in the intersection, up to multiplicative and additive constants. In particular, our inequality reduces to the so-called quasi-factorization of the entropy for commuting algebras, which is a key step in modern proofs of the logarithmic Sobolev inequality for classical lattice spin systems. We also provide estimates on the constants in terms of conditions of clustering of correlations in the setting of quantum lattice spin systems. Along the way, we show the equivalence between conditional expectations arising from Petz recovery maps and those of general Davies semigroups.

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