31 research outputs found
Ferromagnetic phase transition for the spanning-forest model (q \to 0 limit of the Potts model) in three or more dimensions
We present Monte Carlo simulations of the spanning-forest model (q \to 0
limit of the ferromagnetic Potts model) in spatial dimensions d=3,4,5. We show
that, in contrast to the two-dimensional case, the model has a "ferromagnetic"
second-order phase transition at a finite positive value w_c. We present
numerical estimates of w_c and of the thermal and magnetic critical exponents.
We conjecture that the upper critical dimension is 6.Comment: LaTex2e, 4 pages; includes 6 Postscript figures; Version 2 has
expanded title as published in PR
Dynamic critical behavior of the Chayes-Machta-Swendsen-Wang algorithm
We study the dynamic critical behavior of the Chayes-Machta dynamics for the
Fortuin-Kasteleyn random-cluster model, which generalizes the Swendsen-Wang
dynamics for the q-state Potts model to noninteger q, in two and three spatial
dimensions, by Monte Carlo simulation. We show that the Li-Sokal bound z \ge
\alpha/\nu is close to but probably not sharp in d=2, and is far from sharp in
d=3, for all q. The conjecture z \ge \beta/\nu is false (for some values of q)
in both d=2 and d=3.Comment: Revtex4, 4 pages including 4 figure
Cluster simulations of loop models on two-dimensional lattices
We develop cluster algorithms for a broad class of loop models on
two-dimensional lattices, including several standard O(n) loop models at n \ge
1. We show that our algorithm has little or no critical slowing-down when 1 \le
n \le 2. We use this algorithm to investigate the honeycomb-lattice O(n) loop
model, for which we determine several new critical exponents, and a
square-lattice O(n) loop model, for which we obtain new information on the
phase diagram.Comment: LaTex2e, 4 pages; includes 1 table and 2 figures. Totally rewritten
in version 2, with new theory and new data. Version 3 as published in PR
Critical speeding-up in a local dynamics for the random-cluster model
We study the dynamic critical behavior of the local bond-update (Sweeny)
dynamics for the Fortuin-Kasteleyn random-cluster model in dimensions d=2,3, by
Monte Carlo simulation. We show that, for a suitable range of q values, the
global observable S_2 exhibits "critical speeding-up": it decorrelates well on
time scales much less than one sweep, so that the integrated autocorrelation
time tends to zero as the critical point is approached. We also show that the
dynamic critical exponent z_{exp} is very close (possibly equal) to the
rigorous lower bound \alpha/\nu, and quite possibly smaller than the
corresponding exponent for the Chayes-Machta-Swendsen-Wang cluster dynamics.Comment: LaTex2e/revtex4, 4 pages, includes 5 figure
Finite one dimensional impenetrable Bose systems: Occupation numbers
Bosons in the form of ultra cold alkali atoms can be confined to a one
dimensional (1d) domain by the use of harmonic traps. This motivates the study
of the ground state occupations of effective single particle states
, in the theoretical 1d impenetrable Bose gas. Both the system on a
circle and the harmonically trapped system are considered. The and
are the eigenvalues and eigenfunctions respectively of the one body
density matrix. We present a detailed numerical and analytic study of this
problem. Our main results are the explicit scaled forms of the density
matrices, from which it is deduced that for fixed the occupations
are asymptotically proportional to in both the circular
and harmonically trapped cases.Comment: 22 pages, 8 figures (.eps), uses REVTeX
Some geometric critical exponents for percolation and the random-cluster model
We introduce several infinite families of new critical exponents for the
random-cluster model and present scaling arguments relating them to the k-arm
exponents. We then present Monte Carlo simulations confirming these
predictions. These new exponents provide a convenient way to determine k-arm
exponents from Monte Carlo simulations. An understanding of these exponents
also leads to a radically improved implementation of the Sweeny Monte Carlo
algorithm. In addition, our Monte Carlo data allow us to conjecture an exact
expression for the shortest-path fractal dimension d_min in two dimensions:
d_min = (g+2)(g+18)/(32g) where g is the Coulomb-gas coupling, related to the
cluster fugacity q via q = 2 + 2 cos(g\pi/2) with 2 \le g \le 4.Comment: LaTeX2e/Revtex4. Version 2 is completely rewritten to make the
exposition more reader-friendly; it consists of a 4-page main paper
(including 3 figures) and a 2-page EPAPS appendix (given as a single
Postscript file). To appear in Phys Rev
On the Coupling Time of the Heat-Bath Process for the Fortuin–Kasteleyn Random–Cluster Model
We consider the coupling from the past implementation of the random-cluster
heat-bath process, and study its random running time, or coupling time. We
focus on hypercubic lattices embedded on tori, in dimensions one to three, with
cluster fugacity at least one. We make a number of conjectures regarding the
asymptotic behaviour of the coupling time, motivated by rigorous results in one
dimension and Monte Carlo simulations in dimensions two and three. Amongst our
findings, we observe that, for generic parameter values, the distribution of
the appropriately standardized coupling time converges to a Gumbel
distribution, and that the standard deviation of the coupling time is
asymptotic to an explicit universal constant multiple of the relaxation time.
Perhaps surprisingly, we observe these results to hold both off criticality,
where the coupling time closely mimics the coupon collector's problem, and also
at the critical point, provided the cluster fugacity is below the value at
which the transition becomes discontinuous. Finally, we consider analogous
questions for the single-spin Ising heat-bath process
Moments of the Position of the Maximum for GUE Characteristic Polynomials and for Log-Correlated Gaussian Processes
We study three instances of log-correlated processes on the interval: the
logarithm of the Gaussian unitary ensemble (GUE) characteristic polynomial, the
Gaussian log-correlated potential in presence of edge charges, and the
Fractional Brownian motion with Hurst index (fBM0). In previous
collaborations we obtained the probability distribution function (PDF) of the
value of the global minimum (equivalently maximum) for the first two processes,
using the {\it freezing-duality conjecture} (FDC). Here we study the PDF of the
position of the maximum through its moments. Using replica, this requires
calculating moments of the density of eigenvalues in the -Jacobi
ensemble. Using Jack polynomials we obtain an exact and explicit expression for
both positive and negative integer moments for arbitrary and
positive integer in terms of sums over partitions. For positive moments,
this expression agrees with a very recent independent derivation by Mezzadri
and Reynolds. We check our results against a contour integral formula derived
recently by Borodin and Gorin (presented in the Appendix A from these authors).
The duality necessary for the FDC to work is proved, and on our expressions,
found to correspond to exchange of partitions with their dual. Performing the
limit and to negative Dyson index , we obtain the
moments of and give explicit expressions for the lowest ones. Numerical
checks for the GUE polynomials, performed independently by N. Simm, indicate
encouraging agreement. Some results are also obtained for moments in Laguerre,
Hermite-Gaussian, as well as circular and related ensembles. The correlations
of the position and the value of the field at the minimum are also analyzed.Comment: 64 page, 5 figures, with Appendix A written by Alexei Borodin and
Vadim Gorin; The appendix H in the ArXiv version is absent in the published
versio