Ordering and Demixing Transitions in Multicomponent Widom-Rowlinson Models

We use Monte Carlo techniques and analytical methods to study the phase diagram of multicomponent Widom-Rowlinson models on a square lattice: there are M species all with the same fugacity z and a nearest neighbor hard core exclusion between unlike particles. Simulations show that for M between two and six there is a direct transition from the gas phase at z < z_d (M) to a demixed phase consisting mostly of one species at z > z_d (M) while for M \geq 7 there is an intermediate ``crystal phase'' for z lying between z_c(M) and z_d(M). In this phase, which is driven by entropy, particles, independent of species, preferentially occupy one of the sublattices, i.e. spatial symmetry but not particle symmetry is broken. The transition at z_d(M) appears to be first order for M \geq 5 putting it in the Potts model universality class. For large M the transition between the crystalline and demixed phase at z_d(M) can be proven to be first order with z_d(M) \sim M-2 + 1/M + ..., while z_c(M) is argued to behave as \mu_{cr}/M, with \mu_{cr} the value of the fugacity at which the one component hard square lattice gas has a transition, and to be always of the Ising type. Explicit calculations for the Bethe lattice with the coordination number q=4 give results similar to those for the square lattice except that the transition at z_d(M) becomes first order at M>2. This happens for all q, consistent with the model being in the Potts universality class.Comment: 26 pages, 15 postscript figure

Search for gravitational waves from Scorpius X-1 in the second Advanced LIGO observing run with an improved hidden Markov model

We present results from a semicoherent search for continuous gravitational waves from the low-mass x-ray binary Scorpius X-1, using a hidden Markov model (HMM) to track spin wandering. This search improves on previous HMM-based searches of LIGO data by using an improved frequency domain matched filter, the J-statistic, and by analyzing data from Advanced LIGO's second observing run. In the frequency range searched, from 60 to 650 Hz, we find no evidence of gravitational radiation. At 194.6 Hz, the most sensitive search frequency, we report an upper limit on gravitational wave strain (at 95% confidence) of h095%=3.47×10-25 when marginalizing over source inclination angle. This is the most sensitive search for Scorpius X-1, to date, that is specifically designed to be robust in the presence of spin wandering. © 2019 American Physical Society

Phase Coexistence and Slow Mixing for the Hard-Core Model on Z 2

The hard-core model has attracted much attention across several disciplines, representing lattice gases in statistical physics and independent sets in the discrete setting. On nite graphs, we are given a parameter λ, and each independent set I arises with probability proportional to λ |I|. On in nite graphs the Gibbs distribution is de ned as a suitable limit with the correct conditional probabilities. In the in nite setting we are interested in determining when this limit is unique and when there is phase coexistence existence of multiple Gibbs states. In the nite setting, for example on nite regions of the square lattice Z2, we are interested in determining when local Markov chains are rapidly mixing. These problems are believed to be related and it is conjectured that both undergo a phase transition at some critical point λ = λc ≈ 3.79 [1]. It remains open whether there is a single critical point, although it was recently shown that on general graphs of maximum degree ∆, the computational complexity of computing the partition function (namely, the λ-weighted count of independent sets) undergoes a phase transition at the unique well-known critical point λc(T∆) at which the ∆-regular in nite tree T ∆ undergoes a transition from uniqueness to having multiple Gibbs states [25, 27]

Origin and evolution of the Palais-Smale condition in critical point theory

In 1963–64, Palais and Smale have introduced a compactness condition, namely condition (C), on real functions of class C 1 defined on a Riemannian manifold modeled upon a Hilbert space, in order to extend Morse theory to this frame and study nonlinear partial differential equations. This condition and some of its variants have been essential in the development of critical point theory on Banach spaces or Banach manifolds, and are referred as Palais–Smale-type conditions. The paper describes their evolution