250 research outputs found
Spiral model, jamming percolation and glass-jamming transitions
The Spiral Model (SM) corresponds to a new class of kinetically constrained
models introduced in joint works with D.S. Fisher [8,9]. They provide the first
example of finite dimensional models with an ideal glass-jamming transition.
This is due to an underlying jamming percolation transition which has
unconventional features: it is discontinuous (i.e. the percolating cluster is
compact at the transition) and the typical size of the clusters diverges faster
than any power law, leading to a Vogel-Fulcher-like divergence of the
relaxation time. Here we present a detailed physical analysis of SM, see [5]
for rigorous proofs. We also show that our arguments for SM does not need any
modification contrary to recent claims of Jeng and Schwarz [10].Comment: 9 pages, 7 figures, proceedings for StatPhys2
Kinetically constrained spin models on trees
We analyze kinetically constrained 0-1 spin models (KCSM) on rooted and
unrooted trees of finite connectivity. We focus in particular on the class of
Friedrickson-Andersen models FA-jf and on an oriented version of them. These
tree models are particularly relevant in physics literature since some of them
undergo an ergodicity breaking transition with the mixed first-second order
character of the glass transition. Here we first identify the ergodicity regime
and prove that the critical density for FA-jf and OFA-jf models coincide with
that of a suitable bootstrap percolation model. Next we prove for the first
time positivity of the spectral gap in the whole ergodic regime via a novel
argument based on martingales ideas. Finally, we discuss how this new technique
can be generalized to analyze KCSM on the regular lattice .Comment: Published in at http://dx.doi.org/10.1214/12-AAP891 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Mixing length scales of low temperature spin plaquettes models
Plaquette models are short range ferromagnetic spin models that play a key
role in the dynamic facilitation approach to the liquid glass transition. In
this paper we perform a rigorous study of the thermodynamic properties of two
dimensional plaquette models, the square and triangular plaquette models. We
prove that for any positive temperature both models have a unique infinite
volume Gibbs measure with exponentially decaying correlations. We analyse the
scaling of three a priori different static correlation lengths in the small
temperature regime, the mixing, cavity and multispin correlation lengths.
Finally, using the symmetries of the model we determine an exact self
similarity property for the infinite volume Gibbs measure.Comment: 33 pages, 9 figure
Smoothening of Depinning Transitions for Directed Polymers with Quenched Disorder
We consider disordered models of pinning of directed polymers on a defect
line, including (1+1)-dimensional interface wetting models, disordered
Poland--Scheraga models of DNA denaturation and other (1+d)-dimensional
polymers in interaction with columnar defects. We consider also random
copolymers at a selective interface. These models are known to have a
(de)pinning transition at some critical line in the phase diagram. In this work
we prove that, as soon as disorder is present, the transition is at least of
second order: the free energy is differentiable at the critical line, and the
order parameter (contact fraction) vanishes continuously at the transition. On
the other hand, it is known that the corresponding non-disordered models can
have a first order (de)pinning transition, with a jump in the order parameter.
Our results confirm predictions based on the Harris criterion.Comment: 4 pages, 1 figure. Version 2: references added, minor changes made.
To appear on Phys. Rev. Let
Spatial correlations in the relaxation of the Kob-Andersen model
We describe spatio-temporal correlations and heterogeneities in a kinetically
constrained glassy model, the Kob-Andersen model. The kinetic constraints of
the model alone induce the existence of dynamic correlation lengths, that
increase as the density increases, in a way compatible with a
double-exponential law. We characterize in detail the trapping time correlation
length, the cooperativity length, and the distribution of persistent clusters
of particles. This last quantity is related to the typical size of blocked
clusters that slow down the dynamics for a given density.Comment: 7 pages, 6 figures, published version (title has changed
Universality for one-dimensional hierarchical coalescence processes with double and triple merges
We consider one-dimensional hierarchical coalescence processes (in short
HCPs) where two or three neighboring domains can merge. An HCP consists of an
infinite sequence of stochastic coalescence processes: each process occurs in a
different "epoch" and evolves for an infinite time, while the evolutions in
subsequent epochs are linked in such a way that the initial distribution of
epoch coincides with the final distribution of epoch . Inside each
epoch a domain can incorporate one of its neighboring domains or both of them
if its length belongs to a certain epoch-dependent finite range. Assuming that
the distribution at the beginning of the first epoch is described by a renewal
simple point process, we prove limit theorems for the domain length and for the
position of the leftmost point (if any). Our analysis extends the results
obtained in [Ann. Probab. 40 (2012) 1377-1435] to a larger family of models,
including relevant examples from the physics literature [Europhys. Lett. 27
(1994) 175-180, Phys. Rev. E (3) 68 (2003) 031504]. It reveals the presence of
a common abstract structure behind models which are apparently very different,
thus leading to very similar limit theorems. Finally, we give here a full
characterization of the infinitesimal generator for the dynamics inside each
epoch, thus allowing us to describe the time evolution of the expected value of
regular observables in terms of an ordinary differential equation.Comment: Published in at http://dx.doi.org/10.1214/12-AAP917 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On the study of jamming percolation
We investigate kinetically constrained models of glassy transitions, and
determine which model characteristics are crucial in allowing a rigorous proof
that such models have discontinuous transitions with faster than power law
diverging length and time scales. The models we investigate have constraints
similar to that of the knights model, introduced by Toninelli, Biroli, and
Fisher (TBF), but differing neighbor relations. We find that such knights-like
models, otherwise known as models of jamming percolation, need a ``No Parallel
Crossing'' rule for the TBF proof of a glassy transition to be valid.
Furthermore, most knight-like models fail a ``No Perpendicular Crossing''
requirement, and thus need modification to be made rigorous. We also show how
the ``No Parallel Crossing'' requirement can be used to evaluate the provable
glassiness of other correlated percolation models, by looking at models with
more stable directions than the knights model. Finally, we show that the TBF
proof does not generalize in any straightforward fashion for three-dimensional
versions of the knights-like models.Comment: 13 pages, 18 figures; Spiral model does satisfy property
Effective Constraints and Physical Coherent States in Quantum Cosmology: A Numerical Comparison
A cosmological model with a cyclic interpretation is introduced, which is
subject to quantum back-reaction and yet can be treated rather completely by
physical coherent state as well as effective constraint techniques. By this
comparison, the role of quantum back-reaction in quantum cosmology is
unambiguously demonstrated. Also the complementary nature of strengths and
weaknesses of the two procedures is illustrated. Finally, effective constraint
techniques are applied to a more realistic model filled with radiation, where
physical coherent states are not available.Comment: 32 pages, 25 figure
Hierarchical pinning models, quadratic maps and quenched disorder
We consider a hierarchical model of polymer pinning in presence of quenched
disorder, introduced by B. Derrida, V. Hakim and J. Vannimenius in 1992, which
can be re-interpreted as an infinite dimensional dynamical system with random
initial condition (the disorder). It is defined through a recurrence relation
for the law of a random variable {R_n}_{n=1,2,...}, which in absence of
disorder (i.e., when the initial condition is degenerate) reduces to a
particular case of the well-known Logistic Map. The large-n limit of the
sequence of random variables 2^{-n} log R_n, a non-random quantity which is
naturally interpreted as a free energy, plays a central role in our analysis.
The model depends on a parameter alpha>0, related to the geometry of the
hierarchical lattice, and has a phase transition in the sense that the free
energy is positive if the expectation of R_0 is larger than a certain threshold
value, and it is zero otherwise. It was conjectured by Derrida et al. (1992)
that disorder is relevant (respectively, irrelevant or marginally relevant) if
1/2<alpha<1 (respectively, alpha<1/2 or alpha=1/2), in the sense that an
arbitrarily small amount of randomness in the initial condition modifies the
critical point with respect to that of the pure (i.e., non-disordered) model if
alpha is larger or equal to 1/2, but not if alpha is smaller than 1/2. Our main
result is a proof of these conjectures for the case alpha different from 1/2.
We emphasize that for alpha>1/2 we find the correct scaling form (for weak
disorder) of the critical point shift.Comment: 26 pages, 2 figures. v3: Theorem 1.6 improved. To appear on Probab.
Theory Rel. Field
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