9 research outputs found
Asymptotic normality of the size of the giant component in a random hypergraph
Recently, we adapted random walk arguments based on work of Nachmias and
Peres, Martin-L\"of, Karp and Aldous to give a simple proof of the asymptotic
normality of the size of the giant component in the random graph above
the phase transition. Here we show that the same method applies to the
analogous model of random -uniform hypergraphs, establishing asymptotic
normality throughout the (sparse) supercritical regime. Previously, asymptotic
normality was known only towards the two ends of this regime.Comment: 11 page
Branching Processes, and Random-Cluster Measures on Trees
Random-cluster measures on infinite regular trees are studied in conjunction
with a general type of `boundary condition', namely an equivalence relation on
the set of infinite paths of the tree. The uniqueness and non-uniqueness of
random-cluster measures are explored for certain classes of equivalence
relations. In proving uniqueness, the following problem concerning branching
processes is encountered and answered. Consider bond percolation on the
family-tree of a branching process. What is the probability that every
infinite path of , beginning at its root, contains some vertex which is
itself the root of an infinite open sub-tree
Limit Theorems and Coexistence Probabilities for the Curie-Weiss Potts Model with an external field
The Curie-Weiss Potts model is a mean field version of the well-known Potts
model. In this model, the critical line is explicitly
known and corresponds to a first order transition when . In the present
paper we describe the fluctuations of the density vector in the whole domain
and , including the conditional fluctuations
on the critical line and the non-Gaussian fluctuations at the extremity of the
critical line. The probabilities of each of the two thermodynamically stable
states on the critical line are also computed. Similar results are inferred for
the Random-Cluster model on the complete graph.Comment: 17 page
Approximating the partition function of the ferromagnetic Potts model
We provide evidence that it is computationally difficult to approximate the
partition function of the ferromagnetic q-state Potts model when q>2.
Specifically we show that the partition function is hard for the complexity
class #RHPi_1 under approximation-preserving reducibility. Thus, it is as hard
to approximate the partition function as it is to find approximate solutions to
a wide range of counting problems, including that of determining the number of
independent sets in a bipartite graph. Our proof exploits the first order phase
transition of the "random cluster" model, which is a probability distribution
on graphs that is closely related to the q-state Potts model.Comment: Minor correction
Counting connected hypergraphs via the probabilistic method
In 1990 Bender, Canfield and McKay gave an asymptotic formula for the number
of connected graphs on with edges, whenever and the nullity
tend to infinity. Asymptotic formulae for the number of connected
-uniform hypergraphs on with edges and so nullity
were proved by Karo\'nski and \L uczak for the case ,
and Behrisch, Coja-Oghlan and Kang for . Here we prove such a
formula for any fixed, and any satisfying and
as . This leaves open only the (much simpler) case
, which we will consider in future work. ( arXiv:1511.04739 )
Our approach is probabilistic. Let denote the random -uniform
hypergraph on in which each edge is present independently with
probability . Let and be the numbers of vertices and edges in
the largest component of . We prove a local limit theorem giving an
asymptotic formula for the probability that and take any given pair
of values within the `typical' range, for any in the supercritical
regime, i.e., when where
and ; our enumerative result then follows
easily.
Taking as a starting point the recent joint central limit theorem for
and , we use smoothing techniques to show that `nearby' pairs of values
arise with about the same probability, leading to the local limit theorem.
Behrisch et al used similar ideas in a very different way, that does not seem
to work in our setting.
Independently, Sato and Wormald have recently proved the special case ,
with an additional restriction on . They use complementary, more enumerative
methods, which seem to have a more limited scope, but to give additional
information when they do work.Comment: Expanded; asymptotics clarified - no significant mathematical
changes. 67 pages (including appendix