We introduce a new update scheme to systematically improve the efficiency of parallel tempering simulations. We show that, by adapting the number of sweeps between replica exchanges to the canonical autocorrelation time, the average round-trip time of a replica in temperature space can be significantly decreased. The temperatures are not dynamically adjusted as in previous attempts but chosen to yield a 50% exchange rate of adjacent replicas. We illustrate the new algorithm with results for the Ising model in two and the Edwards-Anderson Ising spin glass in three dimensions.

Abstract. -In recent analytical work, Biskup et al. (Europhys. Lett., 60 (2002) 21) studied the behaviour of d-dimensional finite-volume liquid-vapour systems at a fixed excess δN of particles above the ambient gas density. By identifying a dimensionless parameter Δ(δN ) and a universal constant Δc(d), they showed in the limit of large system sizes that for Δ < Δc the excess is absorbed in the background ("evaporated" system), while for Δ > Δc a droplet of the dense phase occurs ("condensed" system). Also the fraction λΔ of excess particles forming the droplet is given explicitly. Furthermore, they argue that the same is true for solid-gas systems. By making use of the well-known equivalence of the lattice-gas picture with the spin-(1/2) Ising model, we performed Monte Carlo simulations of the Ising model with nearest-neighbour couplings on a square lattice with periodic boundary conditions at fixed magnetisation, corresponding to a fixed particles excess. To test the applicability of the analytical results to much smaller, practically accessible system sizes, we measured the largest minority droplet, corresponding to the solid phase, at various system sizes (L = 40, . . . , 640). Using analytic values for the spontaneous magnetisation m0, the susceptibility χ and the Wulff interfacial free energy density τW for the infinite system, we were able to determine λΔ numerically in very good agreement with the theoretical prediction.Introduction. -The formation and dissolution of equilibrium droplets at a first-order phase transition is one of the longstanding problems in statistical mechanics [1]. Quantities of particular interest are the size and free energy of a "critical droplet" that needs to be formed before the decay of the metastable state via homogeneous nucleation can start. For large but finite systems, this is signalised by a cusp in the probability density of the order parameter φ towards the phase-coexistence region as depicted in figs. 1 and 2 for the example of the twodimensional (2D) Ising model, where φ = m is the magnetisation. This "transition point" separates an "evaporated" phase with many very small bubbles of the "wrong" phase around the peak at φ 0 from the "condensed phase" phase, in which a large droplet has formed; for configuration snapshots see fig. 3. The droplet eventually grows further towards φ = 0 until it

The three-dimensional lattice Higgs model with compact U(1) gauge symmetry and unit charge is investigated by means of Monte Carlo simulations. The full model with fluctuating Higgs amplitude is simulated, and both energy as well as topological observables are measured. The data show a Higgs and a confined phase separated by a well-defined phase boundary, which is argued to be caused by proliferating vortices. For fixed gauge coupling, the phase boundary consists of a line of first-order phase transitions at small Higgs self-coupling, ending at a critical point. The phase boundary then continues as a Kertész line across which thermodynamic quantities are non-singular. Symmetry arguments are given to support these findings.

In 2003 Biskup [Commun. Math. Phys. 242, 137 (2003)] gave a rigorous proof for the behavior of equilibrium droplets in the two-dimensional (2D) spin-1/2 Ising model (or, equivalently, a lattice gas of particles) on a finite square lattice of volume V with a given excess delta M identical with M-M 0 of magnetization compared to the spontaneous magnetization M 0=m0V . By identifying a dimensionless parameter Delta(delta M) and a universal constant Delta c , they showed in the limit of large system sizes that for DeltaDelta c a droplet of the minority phase occurs ("condensed" system). By minimizing the free energy of the system, they derived an explicit formula for the fraction lambda(Delta) of excess magnetization forming the droplet. To check the applicability of the asymptotic analytical results to much smaller, practically accessible, system sizes, we performed several Monte Carlo simulations of the 2D Ising model with nearest-neighbor couplings on a square lattice at fixed magnetization M . Thereby, we measured the largest minority droplet, corresponding to the condensed phase in the lattice-gas interpretation, at various system sizes (L=40,80,...,640) . With analytical values for the spontaneous magnetization density m 0 , the susceptibility chi , and the Wulff interfacial free energy density tau W for the infinite system, we were able to determine lambda numerically in very good agreement with the theoretical prediction. Furthermore, we did simulations for the spin-1/2 Ising model on a triangular lattice and with next-nearest-neighbor couplings on a square lattice. Again finding a very good agreement with the analytic formula, we demonstrate the universal aspects of the theory with respect to the underlying lattice and type of interaction. For the case of the next-nearest-neighbor model, where tau W is unknown analytically, we present different methods to obtain it numerically by fitting to the distribution of the magnetization density P(m) .

Analyzing football score data with statistical techniques, we investigate how the highly co-operative nature of the game is reflected in averaged properties such as the distributions of scored goals for the home and away teams. It turns out that in particular the tails of the distributions are not well described by independent Bernoulli trials, but rather well modeled by negative binomial or generalized extreme value distributions. To understand this behavior from first principles, we suggest to modify the Bernoulli random process to include a simple component of self-affirmation which seems to describe the data surprisingly well and allows to interpret the observed deviation from Gaussian statistics. The phenomenological distributions used before can be understood as special cases within this framework. We analyzed historical football score data from many leagues in Europe as well as from international tournaments and found the proposed models to be applicable rather universally. In particular, here we compare men's and women's leagues and the separate German leagues during the cold war times and find some remarkable differences.