Exchange Monte Carlo Method and Application to Spin Glass Simulations

We propose an efficient Monte Carlo algorithm for simulating a ``hardly-relaxing" system, in which many replicas with different temperatures are simultaneously simulated and a virtual process exchanging configurations of these replica is introduced. This exchange process is expected to let the system at low temperatures escape from a local minimum. By using this algorithm the three-dimensional $\pm J$ Ising spin glass model is studied. The ergodicity time in this method is found much smaller than that of the multi-canonical method. In particular the time correlation function almost follows an exponential decay whose relaxation time is comparable to the ergodicity time at low temperatures. It suggests that the system relaxes very rapidly through the exchange process even in the low temperature phase.Comment: 10 pages + uuencoded 5 Postscript figures, REVTe

Continuity for self-destructive percolation in the plane

A few years ago two of us introduced, motivated by the study of certain forest-fireprocesses, the self-destructive percolation model (abbreviated as sdp model). A typical configuration for the sdp model with parameters p and delta is generated in three steps: First we generate a typical configuration for the ordinary percolation model with parameter p. Next, we make all sites in the infinite occupied cluster vacant. Finally, each site that was already vacant in the beginning or made vacant by the above action, becomes occupied with probability delta (independent of the other sites). Let theta(p, delta) be the probability that some specified vertex belongs, in the final configuration, to an infinite occupied cluster. In our earlier paper we stated the conjecture that, for the square lattice and other planar lattices, the function theta has a discontinuity at points of the form (p_c, delta), with delta sufficiently small. We also showed remarkable consequences for the forest-fire models. The conjecture naturally raises the question whether the function theta is continuous outside some region of the above mentioned form. We prove that this is indeed the case. An important ingredient in our proof is a (somewhat stronger form of a) recent ingenious RSW-like percolation result of Bollob\'{a}s and Riordan

Morse theory on spaces of braids and Lagrangian dynamics

In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces. Parabolic flows, a type of one-dimensional lattice dynamics, evolve singular braid diagrams in such a way as to decrease their topological complexity; algebraic lengths decrease monotonically. This topological invariant is derived from a Morse-Conley homotopy index and provides a gloablization of `lap number' techniques used in scalar parabolic PDEs. In the second half of the paper we apply this technology to second order Lagrangians via a discrete formulation of the variational problem. This culminates in a very general forcing theorem for the existence of infinitely many braid classes of closed orbits.Comment: Revised version: numerous changes in exposition. Slight modification of two proofs and one definition; 55 pages, 20 figure

Grundstate Properties of the 3D Ising Spin Glass

We study zero--temperature properties of the 3d Edwards--Anderson Ising spin glass on finite lattices up to size $12^3$. Using multicanonical sampling we generate large numbers of groundstate configurations in thermal equilibrium. Finite size scaling with a zero--temperature scaling exponent $y = 0.74 \pm 0.12$ describes the data well. Alternatively, a descriptions in terms of Parisi mean field behaviour is still possible. The two scenarios give significantly different predictions on lattices of size $\ge 12^3$.Comment: LATEX 9pages,figures upon request ,SCRI-9

Constrained Orthogonal Polynomials

We define sets of orthogonal polynomials satisfying the additional constraint of a vanishing average. These are of interest, for example, for the study of the Hohenberg-Kohn functional for electronic or nucleonic densities and for the study of density fluctuations in centrifuges. We give explicit properties of such polynomial sets, generalizing Laguerre and Legendre polynomials. The nature of the dimension 1 subspace completing such sets is described. A numerical example illustrates the use of such polynomials.Comment: 11 pages, 10 figure

The smallest eigenvalue of Hankel matrices

Let H_N=(s_{n+m}),n,m\le N denote the Hankel matrix of moments of a positive measure with moments of any order. We study the large N behaviour of the smallest eigenvalue lambda_N of H_N. It is proved that lambda_N has exponential decay to zero for any measure with compact support. For general determinate moment problems the decay to 0 of lambda_N can be arbitrarily slow or arbitrarily fast. In the indeterminate case, where lambda_N is known to be bounded below by a positive constant, we prove that the limit of the n'th smallest eigenvalue of H_N for N tending to infinity tends rapidly to infinity with n. The special case of the Stieltjes-Wigert polynomials is discussed

Entropy-based analysis of the number partitioning problem

In this paper we apply the multicanonical method of statistical physics on the number-partitioning problem (NPP). This problem is a basic NP-hard problem from computer science, and can be formulated as a spin-glass problem. We compute the spectral degeneracy, which gives us information about the number of solutions for a given cost $E$ and cardinality $m$. We also study an extension of this problem for $Q$ partitions. We show that a fundamental difference on the spectral degeneracy of the generalized ($Q>2$) NPP exists, which could explain why it is so difficult to find good solutions for this case. The information obtained with the multicanonical method can be very useful on the construction of new algorithms.Comment: 6 pages, 4 figure

A Pulsed Synchrotron for Muon Acceleration at a Neutrino Factory

A 4600 Hz pulsed synchrotron is considered as a means of accelerating cool muons with superconducting RF cavities from 4 to 20 GeV/c for a neutrino factory. Eddy current losses are held to less than a megawatt by the low machine duty cycle plus 100 micron thick grain oriented silicon steel laminations and 250 micron diameter copper wires. Combined function magnets with 20 T/m gradients alternating within single magnets form the lattice. Muon survival is 83%.Comment: 4 pages, 1 figures, LaTeX, 5th International Workshop on Neutrino Factories and Superbeams (NuFact 03), 5-11 Jun 2003, New Yor

Monte Carlo simulation and global optimization without parameters

We propose a new ensemble for Monte Carlo simulations, in which each state is assigned a statistical weight $1/k$, where $k$ is the number of states with smaller or equal energy. This ensemble has robust ergodicity properties and gives significant weight to the ground state, making it effective for hard optimization problems. It can be used to find free energies at all temperatures and picks up aspects of critical behaviour (if present) without any parameter tuning. We test it on the travelling salesperson problem, the Edwards-Anderson spin glass and the triangular antiferromagnet.Comment: 10 pages with 3 Postscript figures, to appear in Phys. Rev. Lett