Ballistic behavior for biased self-avoiding walks

For self-avoiding walks on the d$d$-dimensional cubic lattice defined with a positive bias in one of the coordinate directions, it is proved that the drift in the favored direction is strictly positive.     

Critical Region for Droplet Formation in the Two-Dimensional Ising Model

We study the formation/dissolution of equilibrium droplets in finite systems at parameters corresponding to phase coexistence. Specifically, we consider the 2D Ising model in volumes of size L ² , inverse temperature > _c and overall magnetization conditioned to take the value m L ² –2m v _L , where _c ^–1 is the critical temperature, m =m () is the spontaneous magnetization and v _L is a sequence of positive numbers. We find that the critical scaling for droplet formation/dissolution is when v _L ^3/2 L ^–2 tends to a definite limit. Specifically, we identify a dimensionless parameter , proportional to this limit, a non-trivial critical value _c and a function such that the following holds: For < _c , there are no droplets beyond log L scale, while for > _c , there is a single, Wulff-shaped droplet containing a fraction _c =2/3 of the magnetization deficit and there are no other droplets beyond the scale of log L. Moreover, and are related via a universal equation that apparently is independent of the details of the system.     

Comment on a recent conjectured solution of the three-dimensional Ising model

In a recent paper published in Philosophical Magazine [Z.-D. Zhang, Phil. Mag. 87 (2007) p.5309], the author advances a conjectured solution for various properties of the three-dimensional Ising model. Here, we disprove the conjecture and point out the flaws in the arguments leading to the conjectured expressions.     

Aggregation and intermediate phases in dilute spin systems

We study a variety of dilute annealed lattice spin systems. For site diluted problems with many internal spin states, we uncover a new phase characterized by the occupation and vacancy of staggered sublattices. In cases where the uniform system has a low temperature phase, the staggered states represent an intermediate phase. Furthermore, in many of these cases, we show that (at least part of) the phase boundary separating the low-temperature and staggered phases is a line of phase coexistence-i.e. the transition is first order. We also study the phenomenon of aggregation (phase separation) in bond diluted models. Such transitions are known, trivially, to occur in the large-q Potts models. However, it turns out that phase separation is typical in bond diluted spin systems with many internal states. (In particular, a bond aggregation transition is not tied to a discontinuous transition in the uniform system.) Along the portions of the phase boundary where any of these phenomena occur, the prospects for a Fisher renormalization effect are deemed to be highly unlikely or are ruled out altogether.Partly supported by the NSF grant DMS-93-02023 (L.C.), the grants GAR 202/93/0449 and GAUK 376 (R.K.), and the NSF grant DMS-92-08029 and the Russian Fund of Fundamental Investigations grant 93-01-01470 (S.B.S.).     

On a sharp transition from area law to perimeter law in a system of random surfaces

We introduce and study a phase transition which is associated with the spontaneous formation of infinite surface sheets in a Bernoulli system of random plaquettes. The transition is manifested by a change in the asymptotic behavior of the probability of the formation of a surface, spanning a prescribed loop. As such, this transition offers a generalization of the bond percolation phenomenon. At low plaquette densities, the probability for large loops is shown to decay exponentially with the loops' area, whereas for high densities the decay is by a perimeter law. Furthermore, we show that the two phases of the three dimensional plaquette system are in a precise correspondence with the two phases of the dual system of random bonds. Thus, if a natural conjecture about the phase structure of the bond percolation model is true, then there is a sharp transition in the asymptotic behavior of the surface events. Our analysis incorporates block variables, in terms of which a non-critical system is transformed into one which is close to a trivial, high or low density, fixed point. Stochastic geometric effects like those discussed here play an important role in lattice gauge theories.     

Proof of the local REM conjecture for number partitioning. I: Constant energy scales

In this article we consider the number partitioning problem (NPP ) in the following probabilistic version: Given n numbers X₁,…,X_n drawn i.i.d. from some distribution, one is asked to find the partition into two subsets such that the sum of the numbers in one subset is as close as possible to the sum of the numbers in the other set. In this probabilistic version, the NPP is equivalent to a mean‐field antiferromagnetic Ising spin glass, with spin configurations corresponding to partitions, and the energy of a spin configuration corresponding to the weight difference. Although the energy levels of this model are a priori highly correlated, a surprising recent conjecture of Bauke, Franz, and Mertens asserts that the energy spectrum of number partitioning is locally that of a random energy model (REM): the spacings between nearby energy levels are uncorrelated. More precisely, it was conjectured that the properly scaled energies converge to a Poisson process, and that the spin configurations corresponding to nearby energies are asymptotically uncorrelated. In this article, we prove these two claims, collectively known as the local REM conjecture. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Coexistence of Partially Disordered/Ordered Phases in an Extended Potts Model

We consider a generalization of the standard Potts model in which there are q=r+s states with an interaction that distinguishes the two subspecies. We develop a graphical representation (of the FK type) for the system and show that this representation may be incorporated directly into reflection positivity arguments. Using combinations of these techniques, we establish detailed properties of the phase diagram including the existence of sharp triple points. Whenever relevant, the phases are characterized by the percolation properties of the underlying representation.     

Self-propelled particles with soft-core interactions: patterns, stability, and collapse

Understanding collective properties of driven particle systems is significant for naturally occurring aggregates and because the knowledge gained can be used as building blocks for the design of artificial ones. We model self-propelling biological or artificial individuals interacting through pairwise attractive and repulsive forces. For the first time, we are able to predict stability and morphology of organization starting from the shape of the two-body interaction. We present a coherent theory, based on fundamental statistical mechanics, for all possible phases of collective motion.     

The Phase Diagram of the Quantum Curie-Weiss Model

This paper studies a generalization of the Curie-Weiss model (the Ising model on a complete graph) to quantum mechanics. Using a natural probabilistic representation of this model, we give a complete picture of the phase diagram of the model in the parameters of inverse temperature and transverse field strength. Further analysis computes the critical exponent for the vanishing of the order parameter in the approach to the critical curve and gives useful stability properties for a variational problem associated with the representation.