scholarly journals A spectral condition for spectral gap: fast mixing in high-temperature Ising models

2021 ◽  
Author(s):  
Ronen Eldan ◽  
Frederic Koehler ◽  
Ofer Zeitouni
Keyword(s):  
High Temperature ◽  
Spectral Gap ◽  
Ising Models ◽  
Spectral Condition

scholarly journals Effective high-temperature estimates for intermittent maps

2017 ◽  
Vol 39 (8) ◽  
pp. 2159-2175
Author(s):  
BENOÎT R. KLOECKNER
Keyword(s):  
Perturbation Theory ◽  
High Temperature ◽  
Limit Theorem ◽  
Spectral Gap ◽  
Lipschitz Constant ◽  
Transfer Operator ◽  
Linear Operators ◽  
Suitable Assumption ◽  
Unimodal Map ◽  
Definition Of

Using quantitative perturbation theory for linear operators, we prove a spectral gap for transfer operators of various families of intermittent maps with almost constant potentials (‘high-temperature’ regime). Hölder and bounded $p$-variation potentials are treated, in each case under a suitable assumption on the map, but the method should apply more generally. It is notably proved that for any Pommeau–Manneville map, any potential with Lipschitz constant less than 0.0014 has a transfer operator acting on $\operatorname{Lip}([0,1])$ with a spectral gap; and that for any two-to-one unimodal map, any potential with total variation less than 0.0069 has a transfer operator acting on $\operatorname{BV}([0,1])$ with a spectral gap. We also prove under quite general hypotheses that the classical definition of spectral gap coincides with the formally stronger one used in Giulietti et al [The calculus of thermodynamical formalism. J. Eur. Math. Soc., to appear. Preprint, 2015, arXiv:1508.01297], allowing all results there to be applied under the high-temperature bounds proved here: analyticity of pressure and equilibrium states, central limit theorem, etc.

The Grassmann Representation of Ising Model

1998 ◽  
Vol 12 (20) ◽  
pp. 1995-2003 ◽  
Author(s):  
K. Nojima
Keyword(s):  
High Temperature ◽  
Ising Model ◽  
Ising Models ◽  
Integral Representations ◽  
Dimensional Case ◽  
Partition Functions ◽  
Fermion Field ◽  
Two Dimensional

The integral representations for the partition functions of Ising models are surveyed. The connection with the underlying fermion field in the two-dimensional case is discussed. The relation between the low and the high-temperature expansions is examined.

ONE-PARTICLE SUBSPACE OF THE GLAUBER DYNAMICS GENERATOR FOR CONTINUOUS PARTICLE SYSTEMS

2004 ◽  
Vol 16 (09) ◽  
pp. 1073-1114 ◽  
Author(s):  
YURI KONDRATIEV ◽  
ROBERT MINLOS ◽  
ELENA ZHIZHINA
Keyword(s):  
High Temperature ◽  
Spectral Gap ◽  
Stochastic Dynamics ◽  
Glauber Dynamics ◽  
Particle Systems ◽  
Low Density ◽  
Continuous Particle ◽  
Bounded Smooth ◽  
The One ◽  
Density Regime

We consider a Glauber-type stochastic dynamics of continuous particle systems in ℝd. We construct a one-particle invariant subspace of the generator of this dynamics in the high temperature and low density regime. We prove that under some additional assumptions on the decay of the potential the restriction of the generator on the one-particle subspace is unitary equivalent to the operator of the multiplication by a bounded smooth real-valued function. As a consequence we estimate the spectral gap of the generator and find the second gap between the one-particle branch and the rest of the spectrum.

On the physical content of kinetic Ising models

1976 ◽  
Vol 54 (23) ◽  
pp. 2340-2345 ◽  
Author(s):  
Charles J. Lumsden ◽  
L. E. H. Trainor
Keyword(s):  
High Temperature ◽  
Spin System ◽  
Ising Models ◽  
Temperature Limit ◽  
Arbitrary Spin ◽  
Physical Content ◽  
High Temperature Limit ◽  
Spin Interactions ◽  
Kinetic Ising Models

It is shown that Glauber's methods are appropriate to the high temperature limit for a stochastic spin system and can be applied to models with arbitrary spin interactions in this limit.

High-temperature series expansions for mixed spin-S-spin-S' Ising models

1990 ◽  
Vol 23 (20) ◽  
pp. 4547-4551 ◽  
Author(s):  
G J A Hunter ◽  
R C L Jenkins ◽  
C J Tinsley
Keyword(s):  
High Temperature ◽  
Ising Models ◽  
Temperature Series ◽  
Series Expansions ◽  
Mixed Spin
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