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Results on the correlations of low density classical and quantum Coulomb systems at equilibrium in three dimensions are reviewed. The exponential decay of particle correlations in the classical Coulomb system-Debye-Hückel screening-is compared and contrasted with the quantum case where strong arguments are presented for the absence of exponential screening. Results and techniques for detailed calculations that determine the asymptotic decay of correlations for quantum systems are discussed. Theorems on the existence of molecules in the Saha regime are reviewed. Finally, new combinatoric formulas for the coefficients of Mayer expansions are presented and their role in proofs of results on Debye-Hückel screening is discussed. Typeset using REVT E X-1 CONTENTS

In this paper we study the ground state energy of a classical gas. Our interest centers mainly on Coulomb systems. We obtain some new lower bounds for the energy of a Coulomb gas. As a corollary of our results we can show that a fermionic system with relativistic kinetic energy and Coulomb interaction is stable. More precisely, let HN(~) be the N particle Hamiltonian N HN(cQ =-c~ E ( -A i ) l / 2 + Z [xi--xj1-1-.~.lxi-Rjl-l + ~2 ]Ri-Rj[ -j ,where A~ is the Laptacian in the variable x i s IR 3 and R~, ..., RN are fixed points in IR 3. We show that for sufficiently large ~, independent of N, the Hamiltonian Hu(e) is nonnegative on the space of square integrable functions to(x1, ..., xN), antisymmetric in the variables x~, 1_< i < N .

This paper is concerned with properties of the wave speed for the stochastically perturbed Fisher-Kolmogorov-Petrovsky-Piscunov (FKPP) equation. It was shown in the classical 1937 paper by Kolmogorov, Petrovsky and Piscunov that the large time behavior of the solution to the FKPP equation with Heaviside initial data is a travelling wave. In a seminal 1995 paper Mueller and Sowers proved that this also holds for a stochastically perturbed FKPP equation. The wave speed depends on the strength σ of the noise. In this paper bounds on the asymptotic behavior of the wave speed c(σ) as σ → 0 and σ → ∞ are obtained.

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