[Paper Review] The Posterior metric and the Goodness of Gibbsianness for transforms of Gibbs measures

Christof Kuelske, Alex Akwasi Opoku|ArXiv.org|Nov 23, 2007

Complex Systems and Time Series Analysis10 references5 citations

TL;DR

This paper introduces the posterior metric to analyze the continuity of conditional probabilities in continuous spin systems under local transformations, enabling Dobrushin uniqueness-based estimates without prior metrics on local state spaces. It proves short-time Gibbsianness for time-evolved Gibbs measures in general continuous spin models, even under strong coupling, by decoupling local and spatial dependencies via the posterior metric, with explicit bounds derived using stochastic processes and orthogonal polynomials.

ABSTRACT

We present a general method to derive continuity estimates for conditional probabilities of general (possibly continuous) spin models sub jected to local transformations. Such systems arise in the study of a stochastic time-evolution of Gibbs measures or as noisy observations. We exhibit the minimal necessary structure for such double-layer systems. Assuming no a priori metric on the local state spaces, we define the posterior metric on the local image space. We show that it allows in a natural way to divide the local part of the continuity estimates from the spatial part (which is treated by Dobrushin uniqueness here). We show in the concrete example of the time evolution of rotators on the q-1 dimensional sphere how this method can be used to obtain estimates in terms of the familiar Euclidean metric.

Motivation & Objective

• To develop a general framework for analyzing conditional probability continuity in continuous spin systems subjected to local transformations.
• To establish conditions under which time-evolved Gibbs measures remain Gibbsian, especially in the strong coupling regime.
• To define a posterior metric on the image space that separates local from spatial dependencies in continuity estimates.
• To prove short-time Gibbsianness for general continuous spin models without relying on model-specific structures.
• To extend Dobrushin uniqueness theory to continuous spin systems by introducing a metric-free approach to local regularity.

Proposed method

• Define the posterior metric on the image space of local transformations, enabling local continuity estimates without prior metrics on the original state space.
• Decouple local and spatial dependencies by using the posterior metric to isolate local regularity from spatial mixing via Dobrushin uniqueness.
• Apply the posterior metric to derive bounds on conditional probabilities using stochastic processes, particularly Brownian motion and first-passage times.
• Use orthogonal polynomial expansions (Legendre polynomials) on the sphere to compute integrals over half-spheres, enabling explicit estimates for spherical spins.
• Leverage concentration inequalities and the Gaussian tail bound to control the probability of large deviations in the diffusion process.
• Establish the Dobrushin interdependence matrix using the posterior metric and diameter bounds, ensuring uniqueness and continuity.

Experimental results

Research questions

• RQ1 Can the Gibbs property be preserved for time-evolved Gibbs measures in continuous spin systems under general local dynamics?
• RQ2 How can local continuity of conditional probabilities be estimated without assuming a prior metric on the local state space?
• RQ3 What is the role of the posterior metric in decoupling local and spatial dependencies in continuity estimates?
• RQ4 Under what conditions does the time-evolved measure remain Gibbsian for continuous spins, especially in strong coupling regimes?
• RQ5 How can explicit bounds on the Dobrushin matrix be derived for continuous spin systems using stochastic processes?

Key findings

• The posterior metric enables local continuity estimates independent of the original state space metric, providing a natural framework for analyzing transformed Gibbs measures.
• For the time evolution of rotators on the (q−1)-sphere, the method yields explicit bounds in terms of the Euclidean metric, proving short-time Gibbsianness.
• The probability that a 1D diffusion starting above zero hits zero by time t is bounded by $ \frac{\sqrt{\pi}x}{2\sqrt{t}} $, which controls the decay of conditional probability tails.
• For odd-degree Legendre polynomials on the interval [-1,0], the integral with respect to the invariant measure is explicitly computed as $ (-1)^m \prod_{i=0}^{m} \left( \frac{2i-1}{q+2i-1} \right) $, enabling precise estimates.
• The Dobrushin matrix entries are bounded by $ \bar{C}_{ij} \leq \frac{\rho_{s'}}{2} \exp\left( \frac{1}{2} \sum_{A \supset \{i,j\}} \delta(\Phi_A) \right) L_{ij} $, ensuring uniqueness under weak assumptions.
• The method proves short-time Gibbsianness for general continuous spin systems, even in strong coupling, without requiring cluster expansion or model-specific symmetries.