190 research outputs found
Slow Mixing of Glauber Dynamics for the Six-Vertex Model in the Ordered Phases
The six-vertex model in statistical physics is a weighted generalization of the ice model on Z^2 (i.e., Eulerian orientations) and the zero-temperature three-state Potts model (i.e., proper three-colorings). The phase diagram of the model represents its physical properties and suggests where local Markov chains will be efficient. In this paper, we analyze the mixing time of Glauber dynamics for the six-vertex model in the ordered phases. Specifically, we show that for all Boltzmann weights in the ferroelectric phase, there exist boundary conditions such that local Markov chains require exponential time to converge to equilibrium. This is the first rigorous result bounding the mixing time of Glauber dynamics in the ferroelectric phase. Our analysis demonstrates a fundamental connection between correlated random walks and the dynamics of intersecting lattice path models (or routings). We analyze the Glauber dynamics for the six-vertex model with free boundary conditions in the antiferroelectric phase and significantly extend the region for which local Markov chains are known to be slow mixing. This result relies on a Peierls argument and novel properties of weighted non-backtracking walks
Torpid Mixing of Markov Chains for the Six-vertex Model on Z^2
In this paper, we study the mixing time of two widely used Markov chain algorithms for the six-vertex model, Glauber dynamics and the directed-loop algorithm, on the square lattice Z^2. We prove, for the first time that, on finite regions of the square lattice these Markov chains are torpidly mixing under parameter settings in the ferroelectric phase and the anti-ferroelectric phase
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
Universality classes in nonequilibrium lattice systems
This work is designed to overview our present knowledge about universality
classes occurring in nonequilibrium systems defined on regular lattices. In the
first section I summarize the most important critical exponents, relations and
the field theoretical formalism used in the text. In the second section I
briefly address the question of scaling behavior at first order phase
transitions. In section three I review dynamical extensions of basic static
classes, show the effect of mixing dynamics and the percolation behavior. The
main body of this work is given in section four where genuine, dynamical
universality classes specific to nonequilibrium systems are introduced. In
section five I continue overviewing such nonequilibrium classes but in coupled,
multi-component systems. Most of the known nonequilibrium transition classes
are explored in low dimensions between active and absorbing states of
reaction-diffusion type of systems. However by mapping they can be related to
universal behavior of interface growth models, which I overview in section six.
Finally in section seven I summarize families of absorbing state system
classes, mean-field classes and give an outlook for further directions of
research.Comment: Updated comprehensive review, 62 pages (two column), 29 figs
included. Scheduled for publication in Reviews of Modern Physics in April
200
Universality classes in nonequilibrium lattice systems
This work is designed to overview our present knowledge about universality
classes occurring in nonequilibrium systems defined on regular lattices. In the
first section I summarize the most important critical exponents, relations and
the field theoretical formalism used in the text. In the second section I
briefly address the question of scaling behavior at first order phase
transitions. In section three I review dynamical extensions of basic static
classes, show the effect of mixing dynamics and the percolation behavior. The
main body of this work is given in section four where genuine, dynamical
universality classes specific to nonequilibrium systems are introduced. In
section five I continue overviewing such nonequilibrium classes but in coupled,
multi-component systems. Most of the known nonequilibrium transition classes
are explored in low dimensions between active and absorbing states of
reaction-diffusion type of systems. However by mapping they can be related to
universal behavior of interface growth models, which I overview in section six.
Finally in section seven I summarize families of absorbing state system
classes, mean-field classes and give an outlook for further directions of
research.Comment: Updated comprehensive review, 62 pages (two column), 29 figs
included. Scheduled for publication in Reviews of Modern Physics in April
200
The effective temperature
This review presents the effective temperature notion as defined from the
deviations from the equilibrium fluctuation-dissipation theorem in out of
equilibrium systems with slow dynamics. The thermodynamic meaning of this
quantity is discussed in detail. Analytic, numeric and experimental
measurements are surveyed. Open issues are mentioned.Comment: 58 page
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
Evolutionary games on graphs
Game theory is one of the key paradigms behind many scientific disciplines
from biology to behavioral sciences to economics. In its evolutionary form and
especially when the interacting agents are linked in a specific social network
the underlying solution concepts and methods are very similar to those applied
in non-equilibrium statistical physics. This review gives a tutorial-type
overview of the field for physicists. The first three sections introduce the
necessary background in classical and evolutionary game theory from the basic
definitions to the most important results. The fourth section surveys the
topological complications implied by non-mean-field-type social network
structures in general. The last three sections discuss in detail the dynamic
behavior of three prominent classes of models: the Prisoner's Dilemma, the
Rock-Scissors-Paper game, and Competing Associations. The major theme of the
review is in what sense and how the graph structure of interactions can modify
and enrich the picture of long term behavioral patterns emerging in
evolutionary games.Comment: Review, final version, 133 pages, 65 figure
Combinatorics
Combinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their
properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic
and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization,
Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions.
This is a report on the meeting, containing extended abstracts of the presentations and a summary of the problem session
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