T. R. (Tom) Hurd## Professor of Mathematics |

- J. Dimock & T.R. Hurd, ``Sine-Gordon revisited'',
Annales Henri Poincaré,
2000.

The papers ``A renormalization group analysis of the Kosterlitz-Thouless phase'' and "Construction of the two-dimensional sine-Gordon theory for beta < 8 pi" (Dimock and Hurd 1991 and 1993) solve two complementary open problems, using an early version of the Brydges--Yau constructive method. The first constructed the so--called KT phase of the Coulomb gas up to the conjectured critical temperature T_{KT}=(8\pi)^{-1}. The second constructed the sine--Gordon QFTabove T_KT. In 1996, my coauthor J. Dimock and I discovered a technical flaw in the method which cast both results into uncertainty. It required enormous effort to find improvements to the method which would overcome the error, and restore all of our results. The above paper does all of this.

- D. Brydges, J. Dimock & T.R.
Hurd, ``A non--Gaussian fixed point for phi^4
in 4-\epsilon Dimensions'', Commun. Math. Phys.
**198**, 111-156, 1998.

Wilson, the originator of the constructive renormalization group (RG), proposed this problem in the early '70s as a non-physical model which is of non-renormalizable type (and hence by naive arguments should be inconsistent), but in fact should have a non-trivial infrared fixed point and be constructable. Taking the parameter epsilon > 0 as a small quantity, my two coauthors and I give a complete construction of the model, including the fixed point and associated stable and unstable manifolds in a Banach space of measures, thus solving the problem proposed by Wilson. The result requires a detailed analysis of a single RG step, coupled with a non-trivial extension of the stable manifold theorem from dynamical systems theory.

- D. Brydges, J. Dimock & T.R. Hurd,
``Estimates on renormalization group transformations'', Can. Jour.
Math.
**50**, 756--793, 1998.

This paper gives a complete model independent treatment of the BYDH method suitable for application to scalar field theories for which the interaction is unbounded but stable.

- D.H.U. Marchetti, T.R. Hurd & P.F. da Veiga, ``The1/N expansion as a perturbation around mean field theory: a
one-dimensional Fermi model'', Commun. Math. Phys.
**179**, 623--646, 1996.

The N-component two--dimensional Gross--Neveu (GN) model was proposed in 1974 as one which although massless to all orders in perturbation theory, should nevertheless for large N develop a non--zero mass by spontaneous symmetry breaking. In this paper, we present a complete combinatorial treatment of the one--dimensional model on a lattice. We verify mean-field predictions, such as exponential decay of correlations, for all values of the coupling constant, for large N.

- D. Brydges, J. Dimock and T.R.
Hurd, ``The short distance behaviour of ( phi^4)_3'', Commun. Math.
Phys.
**172**, 143--186, 1995.

This paper was the first to address via the BYDH method a model with a ``large--field'' problem caused by an unbounded interaction potential. The model treated, the ultraviolet phi^4 model in three dimensions, has been solved independently by several earlier methods. Our paper provides a foundation for the further models we have since addressed.

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last updated 25/10/00