Joseph G. Conlon

On asymptotic limits for the quantum Heisenberg model

The authors discuss various asymptotic limits of the classical and quantum Heisenberg model. They give a new proof that the thermodynamic free energy of the quantum model converges to the free energy of the classical model in the limit of large spins. They also obtain Gaussian and free Bose gas limits for the classical and quantum models respectively.

Upper bound on the free energy of the spin 1/2 Heisenberg ferromagnet

The authors obtain an upper bound on the free energy of the spin 1/2 Heisenberg ferromagnet. The zero field bound is, at low temperature, similar to the formula given by the magnon approximation. That is, its functional dependence on temperature is the same but the constant is different.

Random walk representations of the Heisenberg model

We develop random walk representations for the spin-S Heisenberg ferromagnet with nearest neighbor interactions. We show that the spin-S Heisenberg model is a diffusion with local times controlled by the spin-S Ising model. As a consequence, expectations for the Heisenberg model conditioned on zero diffusion are shown to be Ising expectations.

Convergence to Black-Scholes for ergodic volatility models

We study the eeect of stochastic volatility on option prices. In the fast-mean reversion model for stochastic volatility of 5], we show that there is a full asymptotic expansion for the option price, centered at the Black-Scholes price. We show, however, that this price does not converge in a strong sense to Black-Scholes as the mean-reversion rate increases. We also introduce a general (possibly non-Markovian) ergodic model and prove that, assuming decaying correlation between volatility and asset price, the option price strongly converges to Black-Scholes.

A Brownian motion version of the directed polymer problem

Consider a Brownian particle in three dimensions in a random environment. The environment is determined by a potential random in space and time. It is shown that at small noise the large-time behavior of the particle is diffusive. The diffusion constant depends on the environment. This work generalizes previous results for random walk in a random environment. In these results the diffusion constant does not depend on the environment.

Green’s functions for elliptic and parabolic equations with random coefficients II

This paper is concerned with linear parabolic partial differential equations in divergence form and their discrete analogues. It is assumed that the coefficients of the equation are stationary random variables, random in both space and time. The Greens functions for the equations are then random variables. Regularity properties for expectation values of Greens functions are obtained. In particular, it is shown that the expectation value is a continuously differentiable function in the space variable whose derivatives are bounded by the corresponding derivatives of the Greens function for the heat equation. Similar results are obtained for the related finite difference equations. This paper generalises results of a previous paper which considered the case when the coefficients are constant in time but random in space.

Gaussian limit theorems for diffusion processes and an application

Suppose that L=[summation operator]i, j=1daij(x)[not partial differential]2/[not partial differential]xi[not partial differential]xj is uniformly elliptic. We use XL(t) to denote the diffusion associated with L. In this paper we show that, if the dimension of the set is strictly less than d, the random variable converges in distribution to a standard Gaussian random variable. In fact, we also provide rates of convergence. As an application, these results are used to study a problem of a random walk in a random environment.

Uniform convergence of the free energy of the classical Heisenberg model to that of the gaussian model

We show that the free energy of the classical Heisenberg model converges to the free energy of the Gaussian in the low-temperature limit. The limit is uniform as the field is taken to zero.

Green’s function for elliptic systems: existence and Delmotte–Deuschel bounds

This paper is divided into two parts: In the main deterministic part, we prove that for an open domain