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 commonly thought by
mathematicians, that no precise definition that is useful
for rigorous analysis is within sight. However the
renormalization group
(RG), as
pioneered
by Ken Wilson (Nobel prize in Physics, 1982), provides a good
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 the theory of Gaussian
measures on spaces of functions or distributions is the starting point
and it is a good starting point for any problem which is "close to real
Gaussian". This allows a surprising large range of
applications. My
interests in recent years have been in applications to
self-avoiding walk in four dimensions.
It is possible to express self-avoiding walk and other systems in terms
of nearly Gaussian integrals with supersymmetry which can be studied by
RG. My colleague Joel Feldman is also using RG but more in the
context of condensed matter
physics. His work leads to nearly Gaussian integrals with complex
densities. These integrals are not so well understood but they appear
in any problem where time has a direction, for example random walk in a
random environment which is not symmetric.
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Roland Bauerschmidt, David C. Brydges, Gordon Slade, Structural
stability of a dynamical system near a non-hyperbolic fixed point http://arxiv.org/abs/1211.2477
David Brydges and Thomas Spencer, Fluctuation estimates
for sub-quadratic gradient field actions, Journal of Mathematical Physics (Vol.53,
Issue 9, 2012): URL: http://link.aip.org/link/?JMP/53/095216, DOI: 10.1063/1.4747194
D.C. Brydges, A. Dahlqvist and G. Slade.
The strong interaction limit of continuous-time weakly self-avoiding walk.
In
Probability in Complex Physical Systems:
In Honour of Erwin
Bolthausen and Jürgen Gärtner,
eds. J-D. Deuschel et al., Springer Proceedings in Mathematics
11:275--287, (2012)
PDF file
D. Brydges and G. Slade.
Renormalisation group analysis of weakly self-avoiding walk in dimensions four and higher.
Revised April 27, 2010. In Proceedings of the International Congress
of Mathematicians, 2010, eds. R. Bhatia et al., Volume 4, pp. 2232--2257,
World Scientific, (2010).
PDF file
D.C. Brydges, J.Z. Imbrie, G. Slade.
Functional integral representations for self-avoiding walk.
Probability Surveys,
6:34--61, (2009).
PDF file
Lectures at the 2007 Park City
Summer School (Corrections welcomed).
http://www.ams.org/bookstore-getitem/item=PCMS-16
Statistical Mechanics Edited by: Scott Sheffield, Massachusetts Institute of Technology, Cambridge, MA, and Thomas Spencer, Institute for Advanced Study, Princeton, NJ
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