10 research outputs found
Tunneling behavior of Ising and Potts models in the low-temperature regime
We consider the ferromagnetic -state Potts model with zero external field
in a finite volume and assume that the stochastic evolution of this system is
described by a Glauber-type dynamics parametrized by the inverse temperature
. Our analysis concerns the low-temperature regime ,
in which this multi-spin system has stable equilibria, corresponding to the
configurations where all spins are equal. Focusing on grid graphs with various
boundary conditions, we study the tunneling phenomena of the -state Potts
model. More specifically, we describe the asymptotic behavior of the first
hitting times between stable equilibria as in probability,
in expectation, and in distribution and obtain tight bounds on the mixing time
as side-result. In the special case , our results characterize the
tunneling behavior of the Ising model on grid graphs.Comment: 13 figure
High-Dimensional Lipschitz Functions are Typically Flat
A homomorphism height function on the -dimensional torus
is a function taking integer values on the vertices of the torus with
consecutive integers assigned to adjacent vertices. A Lipschitz height function
is defined similarly but may also take equal values on adjacent vertices. In
each model, we consider the uniform distribution over such functions, subject
to boundary conditions. We prove that in high dimensions, with zero boundary
values, a typical function is very flat, having bounded variance at any fixed
vertex and taking at most values with high probability. Our
results extend to any dimension , if is replaced by an
enhanced version of it, the torus for
some fixed . This establishes one side of a conjectured roughening
transition in dimensions. The full transition is established for a class of
tori with non-equal side lengths. We also find that when is taken to
infinity while remains fixed, a typical function takes at most values
with high probability, where for the homomorphism model and for the
Lipschitz model. Suitable generalizations are obtained when grows with .
Our results apply also to the related model of uniform 3-coloring and
establish, for certain boundary conditions, that a uniformly sampled proper
3-coloring of will be nearly constant on either the even or
odd sub-lattice.
Our proofs are based on a combinatorial transformation and on a careful
analysis of the properties of a class of cutsets which we term odd cutsets. For
the Lipschitz model, our results rely also on a bijection of Yadin. This work
generalizes results of Galvin and Kahn, refutes a conjecture of Benjamini,
Yadin and Yehudayoff and answers a question of Benjamini, H\"aggstr\"om and
Mossel.Comment: 63 pages, 5 figures (containing 10 images). Improved introduction and
layout. Minor correction
Three lectures on random proper colorings of
A proper -coloring of a graph is an assignment of one of colors to
each vertex of the graph so that adjacent vertices are colored differently.
Sample uniformly among all proper -colorings of a large discrete cube in the
integer lattice . Does the random coloring obtained exhibit any
large-scale structure? Does it have fast decay of correlations? We discuss
these questions and the way their answers depend on the dimension and the
number of colors . The questions are motivated by statistical physics
(anti-ferromagnetic materials, square ice), combinatorics (proper colorings,
independent sets) and the study of random Lipschitz functions on a lattice. The
discussion introduces a diverse set of tools, useful for this purpose and for
other problems, including spatial mixing, entropy and coupling methods, Gibbs
measures and their classification and refined contour analysis.Comment: 53 pages, 10 figures; Based on lectures given at the workshop on
Random Walks, Random Graphs and Random Media, September 2019, Munich and at
the school Lectures on Probability and Stochastic Processes XIV, December
2019, Delh
Homomorphisms from the torus
We present a detailed probabilistic and structural analysis of the set of
weighted homomorphisms from the discrete torus , where is
even, to any fixed graph: we show that the corresponding probability
distribution on such homomorphisms is close to a distribution defined
constructively as a certain random perturbation of some dominant phase. This
has several consequences, including solutions (in a strong form) to conjectures
of Engbers and Galvin and a conjecture of Kahn and Park. Special cases include
sharp asymptotics for the number of independent sets and the number of proper
-colourings of (so in particular, the discrete hypercube).
We give further applications to the study of height functions and (generalised)
rank functions on the discrete hypercube and disprove a conjecture of Kahn and
Lawrenz. For the proof we combine methods from statistical physics, entropy and
graph containers and exploit isoperimetric and algebraic properties of the
torus.Comment: 84 pages. References adde
Long-range order in discrete spin systems
We establish long-range order for discrete nearest-neighbor spin systems on
satisfying a certain symmetry assumption, when the dimension
is higher than an explicitly described threshold. The results characterize all
periodic, maximal-pressure Gibbs states of the system. The results further
apply in low dimensions provided that the lattice is replaced by
with and
sufficiently high, where is a cycle of even length. Applications
to specific systems are discussed in detail and models for which new results
are provided include the antiferromagnetic Potts model, Lipschitz height
functions, and the hard-core, Widom--Rowlinson and beach models and their
multi-type extensions. We also establish a formula conjectured by Jenssen and
Keevash for the topological pressure in the high-dimensional limit.Comment: 91 pages, 7 figures. This paper is the companion to arXiv:1808.03597.
Established a formula conjectured by Jenssen and Keevash for the topological
pressure in the high-dimensional limi