arXiv subjects

David Brydges

Publications and source records attributed to David Brydges.

3 recordsLinked to original sources

arXiv2014

On the Virial Series for a Gas of Particles with Uniformly Repulsive Pairwise Interaction

The pressure of a gas of particles with a uniformly repulsive pair interaction in a finite container is shown to satisfy (exactly as a formal object) a "viscous" Hamilton-Jacobi (H-J) equation whose solution in power series is recursively given by the variation of constants formula. We investigate the solution of the H-J and of its Legendre transform equation by the Cauchy-Majorant method and provide a lower bound to the radius of convergence on the virial series of the gas which goes beyond the threshold established by Lagrange's inversion formula. A comparison between the behavior of the Mayer and virial coefficients for different regimes of repulsion intensity is also provided.

math-ph↗

arXiv2010

Renormalisation group analysis of weakly self-avoiding walk in dimensions four and higher

We outline a proof, by a rigorous renormalisation group method, that the critical two-point function for continuous-time weakly self-avoiding walk on Z^d decays as |x|^{-(d-2)} in the critical dimension d=4, and also for all d>4.

math.PR↗

arXiv2007

Joint density for the local times of continuous-time Markov chains

We investigate the local times of a continuous-time Markov chain on an arbitrary discrete state space. For fixed finite range of the Markov chain, we derive an explicit formula for the joint density of all local times on the range, at any fixed time. We use standard tools from the theory of stochastic processes and finite-dimensional complex calculus. We apply this formula in the following directions: (1) we derive large deviation upper estimates for the normalized local times beyond the exponential scale, (2) we derive the upper bound in Varadhan's lemma for any measurable functional of the local times, and (3) we derive large deviation upper bounds for continuous-time simple random walk on large subboxes of $\mathbb{Z}^d$ tending to $\mathbb{Z}^d$ as time diverges. We finally discuss the relation of our density formula to the Ray--Knight theorem for continuous-time simple random walk on $\mathbb{Z}$, which is analogous to the well-known Ray--Knight description of Brownian local times.

math.PR↗