545 research outputs found
Liquid-gas and other unusual thermal phase transitions in some large-N magnets
Much insight into the low temperature properties of quantum magnets has been
gained by generalizing them to symmetry groups of order N, and then studying
the large N limit. In this paper we consider an unusual aspect of their finite
temperature behavior--their exhibiting a phase transition between a perfectly
paramagetic state and a paramagnetic state with a finite correlation length at
N = \infty. We analyze this phenomenon in some detail in the large ``spin''
(classical) limit of the SU(N) ferromagnet which is also a lattice
discretization of the CP^{N-1} model. We show that at N = \infty the order of
the transition is governed by lattice connectivity. At finite values of N, the
transition goes away in one or less dimension but survives on many lattices in
two dimensions and higher, for sufficiently large N. The latter conclusion
contradicts a recent conjecture of Sokal and Starinets, yet is consistent with
the known finite temperature behavior of the SU(2) case. We also report closely
related first order paramagnet-ferromagnet transitions at large N and shed
light on a violation of Elitzur's theorem at infinite N via the large q limit
of the q-state Potts model, reformulated as an Ising gauge theory.Comment: 27 pages, 7 figures. Added clarifications requested by a refere
Random walks in a random environment on a strip: a renormalization group approach
We present a real space renormalization group scheme for the problem of
random walks in a random environment on a strip, which includes one-dimensional
random walk in random environment with bounded non-nearest-neighbor jumps. We
show that the model renormalizes to an effective one-dimensional random walk
problem with nearest-neighbor jumps and conclude that Sinai scaling is valid in
the recurrent case, while in the sub-linear transient phase, the displacement
grows as a power of the time.Comment: 9 page
Non-Coexistence of Infinite Clusters in Two-Dimensional Dependent Site Percolation
This paper presents three results on dependent site percolation on the square
lattice. First, there exists no positively associated probability measure on
{0,1}^{Z^2} with the following properties: a) a single infinite 0cluster exists
almost surely, b) at most one infinite 1*cluster exists almost surely, c) some
probabilities regarding 1*clusters are bounded away from zero. Second, we show
that coexistence of an infinite 1*cluster and an infinite 0cluster is almost
surely impossible when the underlying probability measure is ergodic with
respect to translations, positively associated, and satisfies the finite energy
condition. The third result analyses the typical structure of infinite clusters
of both types in the absence of positive association. Namely, under a slightly
sharpened finite energy condition, the existence of infinitely many disjoint
infinite self-avoiding 1*paths follows from the existence of an infinite
1*cluster. The same holds with respect to 0paths and 0clusters.Comment: 17 pages, 1 figur
Transforming fixed-length self-avoiding walks into radial SLE_8/3
We conjecture a relationship between the scaling limit of the fixed-length
ensemble of self-avoiding walks in the upper half plane and radial SLE with
kappa=8/3 in this half plane from 0 to i. The relationship is that if we take a
curve from the fixed-length scaling limit of the SAW, weight it by a suitable
power of the distance to the endpoint of the curve and then apply the conformal
map of the half plane that takes the endpoint to i, then we get the same
probability measure on curves as radial SLE. In addition to a non-rigorous
derivation of this conjecture, we support it with Monte Carlo simulations of
the SAW. Using the conjectured relationship between the SAW and radial SLE, our
simulations give estimates for both the interior and boundary scaling
exponents. The values we obtain are within a few hundredths of a percent of the
conjectured values
Anomalous diffusion in disordered multi-channel systems
We study diffusion of a particle in a system composed of K parallel channels,
where the transition rates within the channels are quenched random variables
whereas the inter-channel transition rate v is homogeneous. A variant of the
strong disorder renormalization group method and Monte Carlo simulations are
used. Generally, we observe anomalous diffusion, where the average distance
travelled by the particle, []_{av}, has a power-law time-dependence
[]_{av} ~ t^{\mu_K(v)}, with a diffusion exponent 0 \le \mu_K(v) \le 1.
In the presence of left-right symmetry of the distribution of random rates, the
recurrent point of the multi-channel system is independent of K, and the
diffusion exponent is found to increase with K and decrease with v. In the
absence of this symmetry, the recurrent point may be shifted with K and the
current can be reversed by varying the lane change rate v.Comment: 16 pages, 7 figure
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School Choice with Consent: An Experiment
Public school choice often yields student assignments that are neither fair nor efficient. The efficiency-adjusted deferred acceptance mechanism (EADAM) allows students to consent to waive priorities that have no effect on their assignments. A burgeoning recent literature places EADAM at the centre of the trade-off between efficiency and fairness in school choice. Meanwhile, the Flemish Ministry of Education has taken the first steps to implement this algorithm in Belgium. We provide the first experimental evidence on the performance of EADAM against the celebrated deferred acceptance mechanism (DA). We find that both efficiency and truth-telling rates are higher under EADAM than under DA, even though EADAM is not strategy-proof. When the priority waiver is enforced, efficiency further increases, while truth-telling rates decrease relative to the EADAM variants where students can dodge the waiver. Our results challenge the importance of strategy-proofness as a prerequisite for truth-telling and portend a new trade-off between efficiency and vulnerability to preference manipulation
Alternative proof for the localization of Sinai's walk
We give an alternative proof of the localization of Sinai's random walk in
random environment under weaker hypothesis than the ones used by Sinai.
Moreover we give estimates that are stronger than the one of Sinai on the
localization neighborhood and on the probability for the random walk to stay
inside this neighborhood
Conformal Invariance in Percolation, Self-Avoiding Walks and Related Problems
Over the years, problems like percolation and self-avoiding walks have
provided important testing grounds for our understanding of the nature of the
critical state. I describe some very recent ideas, as well as some older ones,
which cast light both on these problems themselves and on the quantum field
theories to which they correspond. These ideas come from conformal field
theory, Coulomb gas mappings, and stochastic Loewner evolution.Comment: Plenary talk given at TH-2002, Paris. 21 pages, 9 figure
Ultra-Slow Vacancy-Mediated Tracer Diffusion in Two Dimensions: The Einstein Relation Verified
We study the dynamics of a charged tracer particle (TP) on a two-dimensional
lattice all sites of which except one (a vacancy) are filled with identical
neutral, hard-core particles. The particles move randomly by exchanging their
positions with the vacancy, subject to the hard-core exclusion. In case when
the charged TP experiences a bias due to external electric field ,
(which favors its jumps in the preferential direction), we determine exactly
the limiting probability distribution of the TP position in terms of
appropriate scaling variables and the leading large-N ( being the discrete
time) behavior of the TP mean displacement ; the latter is
shown to obey an anomalous, logarithmic law . On comparing our results with earlier predictions by Brummelhuis
and Hilhorst (J. Stat. Phys. {\bf 53}, 249 (1988)) for the TP diffusivity
in the unbiased case, we infer that the Einstein relation
between the TP diffusivity and the mobility holds in the leading in order, despite
the fact that both and are not constant but vanish as . We also generalize our approach to the situation with very small but
finite vacancy concentration , in which case we find a ballistic-type law
. We demonstrate that here,
again, both and , calculated in the linear in
approximation, do obey the Einstein relation.Comment: 25 pages, one figure, TeX, submitted to J. Stat. Phy
The integrated density of states of the random graph Laplacian
We analyse the density of states of the random graph Laplacian in the
percolating regime. A symmetry argument and knowledge of the density of states
in the nonpercolating regime allows us to isolate the density of states of the
percolating cluster (DSPC) alone, thereby eliminating trivially localised
states due to finite subgraphs. We derive a nonlinear integral equation for the
integrated DSPC and solve it with a population dynamics algorithm. We discuss
the possible existence of a mobility edge and give strong evidence for the
existence of discrete eigenvalues in the whole range of the spectrum.Comment: 4 pages, 1 figure. Supplementary material available at
http://www.theorie.physik.uni-goettingen.de/~aspel/data/spectrum_supplement.pd
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