Spectral Gap Critical Exponent for Glauber Dynamics of Hierarchical Spin Models

Abstract

We develop a renormalisation group approach to deriving the asymptotics of the spectral gap of the generator of Glauber type dynamics of spin systems with strong correlations (at and near a critical point). In our approach, we derive a spectral gap inequality for the measure recursively in terms of spectral gap inequalities for a sequence of renormalised measures. We apply our method to hierarchical versions of the 4-dimensional n-component |ϕ| 4 model at the critical point and its approach from the high temperature side, and of the 2-dimensional Sine-Gordon and the Discrete Gaussian models in the rough phase (Kosterlitz-Thouless phase). For these models, we show that the spectral gap decays polynomially like the spectral gap of the dynamics of a free field (with a logarithmic correction for the |ϕ| 4 model), the scaling limit of these models in equilibrium.

Log‐Sobolev Inequality for the Continuum Sine‐Gordon Model

“…For the latter we expect that the log-Sobolev constant of the lattice ' 4 model or the Ising model in d h 4 (respectively d > 4) scales as u. log u/´(respectively u) as the critical point is approached with distance u 5 0. Again, for the hierarchical ' 4 model, we proved the analogous statement for the spectral gap in [6], and the results of this paper can again be used to improve the latter result to prove also an analogous log-Sobolev inequality; again see Example 2.7.…”

“…For the infrared problem, the microscopic coupling constant is of order 1, and unlikely field configurations play a more important role in understanding the measure (large field problem); see [19,24,25]. We studied the spectral gap for the hierarchical version of the infrared problem in [6]. Using Theorem 2.6 and the estimates proved in [6], the results for the spectral gap stated in [6] can be improved to results about the log-Sobolev constant; see Example 2.7.…”

“…We studied the spectral gap for the hierarchical version of the infrared problem in [6]. Using Theorem 2.6 and the estimates proved in [6], the results for the spectral gap stated in [6] can be improved to results about the log-Sobolev constant; see Example 2.7.…”

“…Assume the measure h 0 has expectation given by (1.12). Let t h 0 for all t > 0, and define the semigroup T s;t h T [6]. By the same methods, one can extend those estimates to noninteger t with t h O. j / for t P .j; j g 1.…”

“…Using Theorem 2.6 instead of [6, theorem 2.1], the theorems for the spectral gap in [6] can thus be extended to analogous ones for the log-Sobolev constant.…”