75 research outputs found
Glauber Dynamics for Ising Model on Convergent Dense Graph Sequences
We study the Glauber dynamics for Ising model on (sequences of) dense graphs. We view the dense graphs through the lens of graphons. For the ferromagnetic Ising model with inverse temperature beta on a convergent sequence of graphs G_n with limit graphon W we show fast mixing of the Glauber dynamics if beta * lambda_1(W) 1 (where lambda_1(W)is the largest eigenvalue of the graphon). We also show that in the case beta * lambda_1(W) = 1 there is insufficient information to determine the mixing time (it can be either fast or slow)
Approximating stationary distributions of fast mixing Glauber dynamics, with applications to exponential random graphs
We provide a general bound on the Wasserstein distance between two arbitrary
distributions of sequences of Bernoulli random variables. The bound is in terms
of a mixing quantity for the Glauber dynamics of one of the sequences, and a
simple expectation of the other. The result is applied to estimate, with
explicit error, expectations of functions of random vectors for some Ising
models and exponential random graphs in "high temperature" regimes.Comment: Ver3: 24 pages, major revision with new results; Ver2: updated
reference; Ver1: 19 pages, 1 figur
A cluster expansion approach to exponential random graph models
The exponential family of random graphs is among the most widely-studied
network models. We show that any exponential random graph model may
alternatively be viewed as a lattice gas model with a finite Banach space norm.
The system may then be treated by cluster expansion methods from statistical
mechanics. In particular, we derive a convergent power series expansion for the
limiting free energy in the case of small parameters. Since the free energy is
the generating function for the expectations of other random variables, this
characterizes the structure and behavior of the limiting network in this
parameter region.Comment: 15 pages, 1 figur
On the dynamics of interfaces in the ferromagnetic XXZ chain under weak perturbations
We study the time evolution of interfaces of the ferromagnetic XXZ chain in a
magnetic field. A scaling limit is introduced where the strength of the
magnetic field tends to zero and the microscopic time to infinity while keeping
their product constant. The leading term and its first correction are
determined and further analyzed in more detail for the case of a uniform
magnetic field.Comment: 20 pages, 2 figures, uses conm-p-l.cls. 1 reference adde
Asymptotically exponential hitting times and metastability: a pathwise approach without reversibility
We study the hitting times of Markov processes to target set , starting
from a reference configuration or its basin of attraction. The
configuration can correspond to the bottom of a (meta)stable well, while
the target could be either a set of saddle (exit) points of the well, or a
set of further (meta)stable configurations. Three types of results are
reported: (1) A general theory is developed, based on the path-wise approach to
metastability, which has three important attributes. First, it is general in
that it does not assume reversibility of the process, does not focus only on
hitting times to rare events and does not assume a particular starting measure.
Second, it relies only on the natural hypothesis that the mean hitting time to
is asymptotically longer than the mean recurrence time to or .
Third, despite its mathematical simplicity, the approach yields precise and
explicit bounds on the corrections to exponentiality. (2) We compare and relate
different metastability conditions proposed in the literature so to eliminate
potential sources of confusion. This is specially relevant for evolutions of
infinite-volume systems, whose treatment depends on whether and how relevant
parameters (temperature, fields) are adjusted. (3) We introduce the notion of
early asymptotic exponential behavior to control time scales asymptotically
smaller than the mean-time scale. This control is particularly relevant for
systems with unbounded state space where nucleations leading to exit from
metastability can happen anywhere in the volume. We provide natural sufficient
conditions on recurrence times for this early exponentiality to hold and show
that it leads to estimations of probability density functions
Droplet Excitations for the Spin-1/2 XXZ Chain with Kink Boundary Conditions
We give a precise definition for excitations consisting of a droplet of size
n in the XXZ chain with various choices of boundary conditions, including kink
boundary conditions and prove that, for each n, the droplet energies converge
to a boundary condition independent value in the thermodynamic limit. We
rigorously compute an explicit formula for this limiting value using the Bethe
ansatz.Comment: 36 pages, figures included as eps file
Exponential Random Graphs and a Generalization of Parking Functions
Random graphs are a powerful tool in the analysis of modern networks. Exponential random graph models provide a framework that allows one to encode desirable subgraph features directly into the probability measure. Using the theory of graph limits pioneered by Borgs et. al. as a foundation, we build upon the work of Chatterjee & Diaconis and Radin & Yin. We add complexity to the previously studied models by considering exponential random graph models with edge-weights coming from a generic distribution satisfying mild assumptions. In particular, we show that a large family of two-parameter, edge-weighted exponential random graphs display a phase transtion and identify the limiting behavior of such graphs in the dual space provided by the Legendre-Fenchel transform.
For finite systems, we analyze the mixing time of exponential random graph models. The mixing time of unweighted exponential random graphs was studied by Bhamidi, Bresler, and Sly. We extend upon the work of Levin, Luczak, and Peres by studying the Glauber dynamics of a certain vertex-weighted exponential random graph model on the complete graph. Specifically, we identify regions of the parameter space where the mixing time is Θ(n log n) and where it is exponentially slow.
Toward the end of this work, we take a drastic turn in a different direction by studying a generalization of parking functions that we call interval parking functions. Parking functions are a classical combinatorial object dating back to the work of Konheim and Weiss in the 1960s. Among other things, we explore the connections that bioutcomes of interval parking functions have to various partial orders on the symmetric group on n letters including the (left) weak order, (strong) Bruhat order, and the bubble-sorting order
Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations
We reconsider the conceptual foundations of the renormalization-group (RG)
formalism, and prove some rigorous theorems on the regularity properties and
possible pathologies of the RG map. Regarding regularity, we show that the RG
map, defined on a suitable space of interactions (= formal Hamiltonians), is
always single-valued and Lipschitz continuous on its domain of definition. This
rules out a recently proposed scenario for the RG description of first-order
phase transitions. On the pathological side, we make rigorous some arguments of
Griffiths, Pearce and Israel, and prove in several cases that the renormalized
measure is not a Gibbs measure for any reasonable interaction. This means that
the RG map is ill-defined, and that the conventional RG description of
first-order phase transitions is not universally valid. For decimation or
Kadanoff transformations applied to the Ising model in dimension ,
these pathologies occur in a full neighborhood of the low-temperature part of the first-order
phase-transition surface. For block-averaging transformations applied to the
Ising model in dimension , the pathologies occur at low temperatures
for arbitrary magnetic-field strength. Pathologies may also occur in the
critical region for Ising models in dimension . We discuss in detail
the distinction between Gibbsian and non-Gibbsian measures, and give a rather
complete catalogue of the known examples. Finally, we discuss the heuristic and
numerical evidence on RG pathologies in the light of our rigorous theorems.Comment: 273 pages including 14 figures, Postscript, See also
ftp.scri.fsu.edu:hep-lat/papers/9210/9210032.ps.
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