12 research outputs found
Finite-size scaling properties of random transverse-field Ising chains : Comparison between canonical and microcanonical ensembles for the disorder
The Random Transverse Field Ising Chain is the simplest disordered model
presenting a quantum phase transition at T=0. We compare analytically its
finite-size scaling properties in two different ensembles for the disorder (i)
the canonical ensemble, where the disorder variables are independent (ii) the
microcanonical ensemble, where there exists a global constraint on the disorder
variables. The observables under study are the surface magnetization, the
correlation of the two surface magnetizations, the gap and the end-to-end
spin-spin correlation for a chain of length . At criticality, each
observable decays typically as in both ensembles, but the
probability distributions of the rescaled variable are different in the two
ensembles, in particular in their asymptotic behaviors. As a consequence, the
dependence in of averaged observables differ in the two ensembles. For
instance, the correlation decays algebraically as 1/L in the canonical
ensemble, but sub-exponentially as in the microcanonical
ensemble. Off criticality, probability distributions of rescaled variables are
governed by the critical exponent in both ensembles, but the following
observables are governed by the exponent in the microcanonical
ensemble, instead of the exponent in the canonical ensemble (a) in the
disordered phase : the averaged surface magnetization, the averaged correlation
of the two surface magnetizations and the averaged end-to-end spin-spin
correlation (b) in the ordered phase : the averaged gap. In conclusion, the
measure of the rare events that dominate various averaged observables can be
very sensitive to the microcanonical constraint.Comment: 24 page
Lingering random walks in random environment on a strip
We consider a recurrent random walk (RW) in random environment (RE) on a
strip. We prove that if the RE is i. i. d. and its distribution is not
supported by an algebraic subsurface in the space of parameters defining the RE
then the RW exhibits the "(log t)-squared" asymptotic behaviour. The
exceptional algebraic subsurface is described by an explicit system of
algebraic equations.
One-dimensional walks with bounded jumps in a RE are treated as a particular
case of the strip model. If the one dimensional RE is i. i. d., then our
approach leads to a complete and constructive classification of possible types
of asymptotic behaviour of recurrent random walks. Namely, the RW exhibits the
asymptotic behaviour if the distribution of the RE is not
supported by a hyperplane in the space of parameters which shall be explicitly
described. And if the support of the RE belongs to this hyperplane then the
corresponding RW is a martingale and its asymptotic behaviour is governed by
the Central Limit Theorem
Random walks and polymers in the presence of quenched disorder
After a general introduction to the field, we describe some recent results
concerning disorder effects on both `random walk models', where the random walk
is a dynamical process generated by local transition rules, and on `polymer
models', where each random walk trajectory representing the configuration of a
polymer chain is associated to a global Boltzmann weight. For random walk
models, we explain, on the specific examples of the Sinai model and of the trap
model, how disorder induces anomalous diffusion, aging behaviours and Golosov
localization, and how these properties can be understood via a strong disorder
renormalization approach. For polymer models, we discuss the critical
properties of various delocalization transitions involving random polymers. We
first summarize some recent progresses in the general theory of random critical
points : thermodynamic observables are not self-averaging at criticality
whenever disorder is relevant, and this lack of self-averaging is directly
related to the probability distribution of pseudo-critical temperatures
over the ensemble of samples of size . We describe the
results of this analysis for the bidimensional wetting and for the
Poland-Scheraga model of DNA denaturation.Comment: 17 pages, Conference Proceedings "Mathematics and Physics", I.H.E.S.,
France, November 200
The near-critical planar FK-Ising model
We study the near-critical FK-Ising model. First, a determination of the
correlation length defined via crossing probabilities is provided. Second, a
phenomenon about the near-critical behavior of FK-Ising is highlighted, which
is completely missing from the case of standard percolation: in any monotone
coupling of FK configurations (e.g., in the one introduced in
[Gri95]), as one raises near , the new edges arrive in a
self-organized way, so that the correlation length is not governed anymore by
the number of pivotal edges at criticality.Comment: 34 pages, 8 figures. This is a streamlined version; the previous one
contains more explanations and additional material on exceptional times in FK
models with general . Furthermore, the statement and proof of Theorem 1.2
have slightly change
Volume growth and heat kernel estimates for the continuum random tree
In this article, we prove global and local (point-wise) volume and heat kernel bounds for the continuum random tree. We demonstrate that there are almost-surely logarithmic global fluctuations and log-logarithmic local fluctuations in the volume of balls of radius r about the leading order polynomial term as r similar to 0. We also show that the on-diagonal part of the heat kernel exhibits corresponding global and local fluctuations as t similar to 0 almost-surely. Finally, we prove that this quenched (almost-sure) behaviour contrasts with the local annealed (averaged over all realisations of the tree) volume and heat kernel behaviour, which is smooth