2,224 research outputs found
Random enriched trees with applications to random graphs
We establish limit theorems that describe the asymptotic local and global
geometric behaviour of random enriched trees considered up to symmetry. We
apply these general results to random unlabelled weighted rooted graphs and
uniform random unlabelled -trees that are rooted at a -clique of
distinguishable vertices. For both models we establish a Gromov--Hausdorff
scaling limit, a Benjamini--Schramm limit, and a local weak limit that
describes the asymptotic shape near the fixed root
Glauber Dynamics on Trees and Hyperbolic Graphs
We study continuous time Glauber dynamics for random configurations with
local constraints (e.g. proper coloring, Ising and Potts models) on finite
graphs with vertices and of bounded degree. We show that the relaxation
time
(defined as the reciprocal of the spectral gap ) for
the dynamics on trees and on planar hyperbolic graphs, is polynomial in .
For these hyperbolic graphs, this yields a general polynomial sampling
algorithm for random configurations. We then show that if the relaxation time
satisfies , then the correlation coefficient, and the
mutual information, between any local function (which depends only on the
configuration in a fixed window) and the boundary conditions, decays
exponentially in the distance between the window and the boundary. For the
Ising model on a regular tree, this condition is sharp.Comment: To appear in Probability Theory and Related Field
Broadcasting on Random Directed Acyclic Graphs
We study a generalization of the well-known model of broadcasting on trees.
Consider a directed acyclic graph (DAG) with a unique source vertex , and
suppose all other vertices have indegree . Let the vertices at
distance from be called layer . At layer , is given a random
bit. At layer , each vertex receives bits from its parents in
layer , which are transmitted along independent binary symmetric channel
edges, and combines them using a -ary Boolean processing function. The goal
is to reconstruct with probability of error bounded away from using
the values of all vertices at an arbitrarily deep layer. This question is
closely related to models of reliable computation and storage, and information
flow in biological networks.
In this paper, we analyze randomly constructed DAGs, for which we show that
broadcasting is only possible if the noise level is below a certain degree and
function dependent critical threshold. For , and random DAGs with
layer sizes and majority processing functions, we identify the
critical threshold. For , we establish a similar result for NAND
processing functions. We also prove a partial converse for odd
illustrating that the identified thresholds are impossible to improve by
selecting different processing functions if the decoder is restricted to using
a single vertex.
Finally, for any noise level, we construct explicit DAGs (using expander
graphs) with bounded degree and layer sizes admitting
reconstruction. In particular, we show that such DAGs can be generated in
deterministic quasi-polynomial time or randomized polylogarithmic time in the
depth. These results portray a doubly-exponential advantage for storing a bit
in DAGs compared to trees, where but layer sizes must grow exponentially
with depth in order to enable broadcasting.Comment: 33 pages, double column format. arXiv admin note: text overlap with
arXiv:1803.0752
Phase ordering after a deep quench: the stochastic Ising and hard core gas models on a tree
Consider a low temperature stochastic Ising model in the phase coexistence
regime with Markov semigroup . A fundamental and still largely open
problem is the understanding of the long time behavior of \d_\h P_t when the
initial configuration \h is sampled from a highly disordered state
(e.g. a product Bernoulli measure or a high temperature Gibbs measure).
Exploiting recent progresses in the analysis of the mixing time of Monte Carlo
Markov chains for discrete spin models on a regular -ary tree \Tree^b, we
tackle the above problem for the Ising and hard core gas (independent sets)
models on \Tree^b. If is a biased product Bernoulli law then, under
various assumptions on the bias and on the thermodynamic parameters, we prove
-almost sure weak convergence of \d_\h P_t to an extremal Gibbs measure
(pure phase) and show that the limit is approached at least as fast as a
stretched exponential of the time . In the context of randomized algorithms
and if one considers the Glauber dynamics on a large, finite tree, our results
prove fast local relaxation to equilibrium on time scales much smaller than the
true mixing time, provided that the starting point of the chain is not taken as
the worst one but it is rather sampled from a suitable distribution.Comment: 35 page
A law of the iterated logarithm for iterated random walks, with application to random recursive trees
Consider a Crump-Mode-Jagers process generated by an increasing random walk
whose increments have finite second moment. Let be the number of
individuals in generation born in the time interval .
We prove a law of the iterated logarithm for with fixed , as . As a consequence, we derive a law of the iterated logarithm for the
number of vertices at a fixed level in a random recursive tree, as the
number of vertices goes to .Comment: 16 page
Decay of Correlations for the Hardcore Model on the -regular Random Graph
A key insight from statistical physics about spin systems on random graphs is
the central role played by Gibbs measures on trees. We determine the local weak
limit of the hardcore model on random regular graphs asymptotically until just
below its condensation threshold, showing that it converges in probability
locally in a strong sense to the free boundary condition Gibbs measure on the
tree. As a consequence we show that the reconstruction threshold on the random
graph, indicative of the onset of point to set spatial correlations, is equal
to the reconstruction threshold on the -regular tree for which we determine
precise asymptotics. We expect that our methods will generalize to a wide range
of spin systems for which the second moment method holds.Comment: 39 pages, 5 figures. arXiv admin note: text overlap with
arXiv:1004.353
On the freezing of variables in random constraint satisfaction problems
The set of solutions of random constraint satisfaction problems (zero energy
groundstates of mean-field diluted spin glasses) undergoes several structural
phase transitions as the amount of constraints is increased. This set first
breaks down into a large number of well separated clusters. At the freezing
transition, which is in general distinct from the clustering one, some
variables (spins) take the same value in all solutions of a given cluster. In
this paper we study the critical behavior around the freezing transition, which
appears in the unfrozen phase as the divergence of the sizes of the
rearrangements induced in response to the modification of a variable. The
formalism is developed on generic constraint satisfaction problems and applied
in particular to the random satisfiability of boolean formulas and to the
coloring of random graphs. The computation is first performed in random tree
ensembles, for which we underline a connection with percolation models and with
the reconstruction problem of information theory. The validity of these results
for the original random ensembles is then discussed in the framework of the
cavity method.Comment: 32 pages, 7 figure
Integrals of motion in the Many-Body localized phase
We construct a complete set of quasi-local integrals of motion for the
many-body localized phase of interacting fermions in a disordered potential.
The integrals of motion can be chosen to have binary spectrum , thus
constituting exact quasiparticle occupation number operators for the Fermi
insulator. We map the problem onto a non-Hermitian hopping problem on a lattice
in operator space. We show how the integrals of motion can be built, under
certain approximations, as a convergent series in the interaction strength. An
estimate of its radius of convergence is given, which also provides an estimate
for the many-body localization-delocalization transition. Finally, we discuss
how the properties of the operator expansion for the integrals of motion imply
the presence or absence of a finite temperature transition.Comment: 65 pages, 12 figures. Corrected typos, added reference
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