Correlation Inequalities for Spin Glasses

We prove a correlation type inequality for spin systems with quenched symmetric random interactions. This gives monotonicity of the pressure with respect to the strength of the interaction for a class of spin glass models. Consequences include existence of the thermodynamic limit for the pressure and bounds on the surface pressure. We also describe other conjectured inequalities for such systems

Hydrodynamic Limit of Brownian Particles Interacting with Short and Long Range Forces

We investigate the time evolution of a model system of interacting particles, moving in a $d$-dimensional torus. The microscopic dynamics are first order in time with velocities set equal to the negative gradient of a potential energy term $\Psi$ plus independent Brownian motions: $\Psi$ is the sum of pair potentials, $V(r)+\gamma^d J(\gamma r)$, the second term has the form of a Kac potential with inverse range $\gamma$. Using diffusive hydrodynamical scaling (spatial scale $\gamma^{-1}$, temporal scale $\gamma^{-2}$) we obtain, in the limit $\gamma\downarrow 0$, a diffusive type integro-differential equation describing the time evolution of the macroscopic density profile.Comment: 37 pages, in TeX (compile twice), to appear on J. Stat. Phys., e-mail addresses: [email protected], [email protected]

Note on a diffraction-amplification problem

We investigate the solution of the equation \partial_t E(x,t)-iD\partial_x^2 E(x,t)= \lambda |S(x,t)|^2 E(x,t)$, for x in a circle and S(x,t) a Gaussian stochastic field with a covariance of a particular form. It is shown that the coupling \lambda_c at which diverges for t>=1 (in suitable units), is always less or equal for D>0 than D=0.Comment: REVTeX file, 8 pages, submitted to Journal of Physics