Interests:  Statistical Mechanics, Quantum Field Theory, Functional Integration, Probability

A large part of theoretical physics is built around the “functional integral” formulation of quantum field theory.  These functional integrals are defined in the sense of formal power series (renormalised perturbation theory).  It is widely, but wrongly believed, by mathematicians, that no precise definition that is useful for rigorous analysis is within sight.  The renormalization group (RG), as pioneered by Ken Wilson (Nobel prize in Physics, 1982),  provides a clear roadmap for defining functional integrals and studying the remarkable  range of phenomena contained within them, in particular, renormalisation, scaling limits and the phase transitions of statistical mechanics. In these cases one can work with integrals based on measures on spaces of functions as opposed to complex valued "measures" on spaces of functions.  The complex valued case (Feynman functional integrals) is indeed further toward the horizon of difficulty.  Without facing the difficulties of the complex valued case, there is already an enormous range of possible applications.  My interests in  recent years have been in applications to  self-avoiding walk in four dimensions.  Functional integrals  combine with supersymmetry to generate combinatoric identities so whenever I need a rest from the RG I like to think about that aspect as well. The papers below are a mixture of themes involving supersymmetry and analysis by RG.    My colleague Joel Feldman is using closely related ideas to prove results in the context of condensed matter physics.

Combinatoric Result

A. Abdesselam and D. C. Brydges, Cramer's Rule and Loop Ensembles
This is a review of a result of G.X. Viennot which we think is important for statistical mechanics.

Recent papers on the Renormalization Group

Lectures at the 2007 Park City Summer School   To be published by the AMS. ( Corrections welcomed).

Brydges, David and Talarczyk, Anna,  Finite range decompositions of positive-definite functions,
Journal of Functional Analysis, Volume 236, Issue 2, 15 July 2006, Pages 682-711
http://dx.doi.org/10.1016/j.jfa.2006.03.008

David C.Brydges, G.Guadagni, P.K.Mitter
Finite range Decomposition of Gaussian Processes JSP  2004, 115,  pages {415--449},
 
D. C. Brydges, P. K. Mitter, B. Scoppola.  CRITICAL (Phi^{4}_{3,/epsilon})  Communications in Mathematical Physics, 240, 2003, pages  281--327



 

 

On Self Avoiding Walk and related problems (Applications of functional integration and supersymmetry)

Lectures sponsored by PIMS  http://www.math.ubc.ca/~db5d/Seminars/PIMSLectures2001/lectures.pdf

David C. Brydges, John Z. Imbrie.  Green's Function for a Hierarchical Self-Avoiding Walk in Four Dimensions,  Communications in Mathematical Physics,

239, 2003, pages 549--584


David C. Brydges, John Z. Imbrie.  End-to-end Distance from the Green's Function for a Hierarchical Self-Avoiding Walk
in Four Dimension Communications in Mathematical Physics,  239, 2003, pages 523--547


David C. Brydges, John Z. Imbrie.  Dimensional Reduction Formulas for Branched Polymer Correlation Functions,  Journal of Statistical Physics,
110,  2003, pages 503--518

David C. Brydges, John Z. Imbrie.  Branched Polymers and Dimensional Reduction,     Annals of Mathematics,

2003,  158,  pages 1019­-103


 
On Coulomb Systems (Application of Gaussian Functional Integration)
David C. Brydges, Ph. A. Martin.    Coulomb systems at low density, Journal of Statistical Physics,  96, 1999, 1163--1330