Frederic Green

Mean field analysis of SU(N) deconfining transitions in the presence of dynamical quarks

Abstract The deconfining transitions of SU( N ) lattice gauge theories, both with and without quarks, are studied using strong coupling techniques combined with a mean field analysis. In the pure gauge sector our analysis suggests first-order transitions for N ⩾ 3 and a second-order transition only for N = 2. Quarks are incorporated via an effective external field h related to the Wilson hopping parameter. For N = 2 the transition disappears for arbitrarily small h , whereas for finite N ⩾ 3 it disappears above non-zero critical field h c . h c approaches zero as N → ∞ even though the pure gauge sector transition remains first order. Our results for h c in the SU(3) case agree well with recent Monte Carlo simulations.

On the correlation of symmetric functions

Thecorrelation between two Boolean functions ofn inputs is defined as the number of times the functions agree minus the number of times they disagree, all divided by 2n. In this paper we compute, in closed form, the correlation between any twosymmetric Boolean functions. As a consequence of our main result, we get that every symmetric Boolean function having an odd period has anexponentially small correlation (inn) with the parity function. This improves a result of Smolensky [12] restricted to symmetric Boolean functions: the correlation between parity and any circuit consisting of a Modq gate over AND gates of small fan-in, whereq is odd and the function computed by the sum of the AND gates is symmetric, is bounded by 2−Ω(n).In addition, we find that for a large class of symmetric functions the correlation with parity isidentically zero for infinitely manyn. We characterize exactly those symmetric Boolean functions having this property.

Chiral models: Their implication for gauge theories and large N

Abstract Two-dimensional chiral models are investigated from a strong coupling viewpoint, with the ultimate objective of understanding the phase structure of gauge theories. Predictions for the average link and beta function are obtained. The abelian case agrees well with previous work. Strong evidence for a phase transition in the U( N ) cases is found, in contrast to the SU(2) and SU(3) theories. Large N character expansion techniques are developed and give strong evidence for a large N phase transition. Our conclusions can be applied qualitatively to gauge theories, with one difference: instantons are necessary in QCD for connecting weak with strong coupling.

On the power of deterministic reductions to C=P

AbstractThe counting class C=P, which captures the notion of “exact counting”, while extremely powerful under various nondeterministic reductions, is quite weak under polynomial-time deterministic reductions. We discuss the analogies between NP and co-C=P, which allow us to derive many interesting results for such deterministic reductions to co-C=P. We exploit these results to obtain some interesting oracle separations. Most importantly, we show that there exists an oracleA such that

The correlation between parity and quadratic polynomials mod 3

\oplus P^A \nsubseteq P^{C_ = P^A }

The large-N phase transition in the U(N) lattice gauge theory

and

Strong coupling expansions for the string tension at finite temperature

BPP^A \nsubseteq P^{C_ = P^A }