2,473 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
Physics of Skiing: The Ideal-Carving Equation and Its Applications
Ideal carving occurs when a snowboarder or skier, equipped with a snowboard
or carving skis, describes a perfect carved turn in which the edges of the ski
alone, not the ski surface, describe the trajectory followed by the skier,
without any slipping or skidding. In this article, we derive the
"ideal-carving" equation which describes the physics of a carved turn under
ideal conditions. The laws of Newtonian classical mechanics are applied. The
parameters of the ideal-carving equation are the inclination of the ski slope,
the acceleration of gravity, and the sidecut radius of the ski. The variables
of the ideal-carving equation are the velocity of the skier, the angle between
the trajectory of the skier and the horizontal, and the instantaneous curvature
radius of the skier's trajectory. Relations between the slope inclination and
the velocity range suited for nearly ideal carving are discussed, as well as
implications for the design of carving skis and snowboards.Comment: 13 pages, 9 figures, LaTeX; to appear in Can. J. Phy
Analyzing Boltzmann Samplers for Bose-Einstein Condensates with Dirichlet Generating Functions
Boltzmann sampling is commonly used to uniformly sample objects of a
particular size from large combinatorial sets. For this technique to be
effective, one needs to prove that (1) the sampling procedure is efficient and
(2) objects of the desired size are generated with sufficiently high
probability. We use this approach to give a provably efficient sampling
algorithm for a class of weighted integer partitions related to Bose-Einstein
condensation from statistical physics. Our sampling algorithm is a
probabilistic interpretation of the ordinary generating function for these
objects, derived from the symbolic method of analytic combinatorics. Using the
Khintchine-Meinardus probabilistic method to bound the rejection rate of our
Boltzmann sampler through singularity analysis of Dirichlet generating
functions, we offer an alternative approach to analyze Boltzmann samplers for
objects with multiplicative structure.Comment: 20 pages, 1 figur
Non-monotone Submodular Maximization with Nearly Optimal Adaptivity and Query Complexity
Submodular maximization is a general optimization problem with a wide range
of applications in machine learning (e.g., active learning, clustering, and
feature selection). In large-scale optimization, the parallel running time of
an algorithm is governed by its adaptivity, which measures the number of
sequential rounds needed if the algorithm can execute polynomially-many
independent oracle queries in parallel. While low adaptivity is ideal, it is
not sufficient for an algorithm to be efficient in practice---there are many
applications of distributed submodular optimization where the number of
function evaluations becomes prohibitively expensive. Motivated by these
applications, we study the adaptivity and query complexity of submodular
maximization. In this paper, we give the first constant-factor approximation
algorithm for maximizing a non-monotone submodular function subject to a
cardinality constraint that runs in adaptive rounds and makes
oracle queries in expectation. In our empirical study, we use
three real-world applications to compare our algorithm with several benchmarks
for non-monotone submodular maximization. The results demonstrate that our
algorithm finds competitive solutions using significantly fewer rounds and
queries.Comment: 12 pages, 8 figure
Nearly Tight Bounds for Sandpile Transience on the Grid
We use techniques from the theory of electrical networks to give nearly tight
bounds for the transience class of the Abelian sandpile model on the
two-dimensional grid up to polylogarithmic factors. The Abelian sandpile model
is a discrete process on graphs that is intimately related to the phenomenon of
self-organized criticality. In this process, vertices receive grains of sand,
and once the number of grains exceeds their degree, they topple by sending
grains to their neighbors. The transience class of a model is the maximum
number of grains that can be added to the system before it necessarily reaches
its steady-state behavior or, equivalently, a recurrent state. Through a more
refined and global analysis of electrical potentials and random walks, we give
an upper bound and an lower bound for the
transience class of the grid. Our methods naturally extend to
-sized -dimensional grids to give upper
bounds and lower bounds.Comment: 36 pages, 4 figure
Approximately Sampling Elements with Fixed Rank in Graded Posets
Graded posets frequently arise throughout combinatorics, where it is natural
to try to count the number of elements of a fixed rank. These counting problems
are often -complete, so we consider approximation algorithms for
counting and uniform sampling. We show that for certain classes of posets,
biased Markov chains that walk along edges of their Hasse diagrams allow us to
approximately generate samples with any fixed rank in expected polynomial time.
Our arguments do not rely on the typical proofs of log-concavity, which are
used to construct a stationary distribution with a specific mode in order to
give a lower bound on the probability of outputting an element of the desired
rank. Instead, we infer this directly from bounds on the mixing time of the
chains through a method we call .
A noteworthy application of our method is sampling restricted classes of
integer partitions of . We give the first provably efficient Markov chain
algorithm to uniformly sample integer partitions of from general restricted
classes. Several observations allow us to improve the efficiency of this chain
to require space, and for unrestricted integer partitions,
expected time. Related applications include sampling permutations
with a fixed number of inversions and lozenge tilings on the triangular lattice
with a fixed average height.Comment: 23 pages, 12 figure
- …