Conditional Expectations and Renormalization

TL;DR

This paper explores the connection between optimal prediction and renormalization group methods for thermal systems, developing a framework to compute reduced models and critical parameters via series expansion and Monte Carlo evaluation.

Contribution

It establishes a formal relation between optimal prediction and renormalization group techniques, enabling systematic derivation of reduced models and parameter flows for complex systems.

Findings

Derived renormalization parameter flows for spin systems

Calculated critical temperature and magnetization using reduced models

Demonstrated the method with simple spin system examples

Abstract

In optimal prediction methods one estimates the future behavior of underresolved systems by solving reduced systems of equations for expectations conditioned by partial data; renormalization group methods reduce the number of variables in complex systems through integration of unwanted scales. We establish the relation between these methods for systems in thermal equilibrium, and use this relation to find renormalization parameter flows and the coefficients in reduced systems by expanding conditional expectations in series and evaluating the coefficients by Monte-Carlo. We illustrate the construction by finding parameter flows for simple spin systems and then using the renormalized (=reduced) systems to calculate the critical temperature and the magnetization.