Nicholas Crawford

Simple random walk on long-range percolation clusters II: Scaling limits

Supported in part at the Technion by a Landau fellowship. Supported in part by an Alfred Sloan Fellowship in Mathematics

The Phase Diagram of the Quantum Curie-Weiss Model

This paper studies a generalization of the Curie-Weiss model (the Ising model on a complete graph) to quantum mechanics. Using a natural probabilistic representation of this model, we give a complete picture of the phase diagram of the model in the parameters of inverse temperature and transverse field strength. Further analysis computes the critical exponent for the vanishing of the order parameter in the approach to the critical curve and gives useful stability properties for a variational problem associated with the representation.

Emptiness Formation Probability

We present rigorous upper and lower bounds on the emptiness formation probability for the ground state of a spin-1/2 Heisenberg XXZ quantum spin system. For a d-dimensional system we find a rate of decay of the order

On Random Field Induced Ordering in the Classical XY Model

{\exp(-c L^{d+1})}

The interaction between multioverlaps in the high temperature phase of the Sherrington–Kirkpatrick spin glass

exp(-cLd+1) where L is the sidelength of the box in which we ask for the emptiness formation event to occur. In the

Random Current Representation for Transverse Field Ising Model

{d=1}