23 research outputs found
Scaling limits of random skew plane partitions with arbitrarily sloped back walls
The paper studies scaling limits of random skew plane partitions confined to
a box when the inner shapes converge uniformly to a piecewise linear function V
of arbitrary slopes in [-1,1]. It is shown that the correlation kernels in the
bulk are given by the incomplete Beta kernel, as expected. As a consequence it
is established that the local correlation functions in the scaling limit do not
depend on the particular sequence of discrete inner shapes that converge to V.
A detailed analysis of the correlation kernels at the top of the limit shape
and of the frozen boundary is given. It is shown that depending on the slope of
the linear section of the back wall, the system exhibits behavior observed in
either [OR2] or [BMRT].Comment: 29 pages. Version 2: Several sections and proofs were improved and
completely rewritten. These include Sections 2.2.2,2.2.4 and 2.2.5, Lemmas
2.3 and 4.2, and Proposition 4.1. Section 1.1.3 was added. This version is to
be published in Comm. Math. Phy
Random skew plane partitions with a piecewise periodic back wall
Random skew plane partitions of large size distributed according to an
appropriately scaled Schur process develop limit shapes. In the present work we
consider the limit of large random skew plane partitions where the inner
boundary approaches a piecewise linear curve with non-lattice slopes,
describing the limit shape and the local fluctuations in various regions. This
analysis is fairly similar to that in [OR2], but we do find some new behavior.
For instance, the boundary of the limit shape is now a single smooth (not
algebraic) curve, whereas the boundary in [OR2] is singular. We also observe
the bead process introduced in [B] appearing in the asymptotics at the top of
the limit shape.Comment: 24 pages. This version to appear in Annales Henri Poincar
Gibbs Ensembles of Nonintersecting Paths
We consider a family of determinantal random point processes on the
two-dimensional lattice and prove that members of our family can be interpreted
as a kind of Gibbs ensembles of nonintersecting paths. Examples include
probability measures on lozenge and domino tilings of the plane, some of which
are non-translation-invariant.
The correlation kernels of our processes can be viewed as extensions of the
discrete sine kernel, and we show that the Gibbs property is a consequence of
simple linear relations satisfied by these kernels. The processes depend on
infinitely many parameters, which are closely related to parametrization of
totally positive Toeplitz matrices.Comment: 6 figure
Brane Tilings and Exceptional Collections
Both brane tilings and exceptional collections are useful tools for
describing the low energy gauge theory on a stack of D3-branes probing a
Calabi-Yau singularity. We provide a dictionary that translates between these
two heretofore unconnected languages. Given a brane tiling, we compute an
exceptional collection of line bundles associated to the base of the
non-compact Calabi-Yau threefold. Given an exceptional collection, we derive
the periodic quiver of the gauge theory which is the graph theoretic dual of
the brane tiling. Our results give new insight to the construction of quiver
theories and their relation to geometry.Comment: 46 pages, 37 figures, JHEP3; v2: reference added, figure 13 correcte
The arctic curve of the domain-wall six-vertex model
The problem of the form of the `arctic' curve of the six-vertex model with
domain wall boundary conditions in its disordered regime is addressed. It is
well-known that in the scaling limit the model exhibits phase-separation, with
regions of order and disorder sharply separated by a smooth curve, called the
arctic curve. To find this curve, we study a multiple integral representation
for the emptiness formation probability, a correlation function devised to
detect spatial transition from order to disorder. We conjecture that the arctic
curve, for arbitrary choice of the vertex weights, can be characterized by the
condition of condensation of almost all roots of the corresponding saddle-point
equations at the same, known, value. In explicit calculations we restrict to
the disordered regime for which we have been able to compute the scaling limit
of certain generating function entering the saddle-point equations. The arctic
curve is obtained in parametric form and appears to be a non-algebraic curve in
general; it turns into an algebraic one in the so-called root-of-unity cases.
The arctic curve is also discussed in application to the limit shape of
-enumerated (with ) large alternating sign matrices. In
particular, as the limit shape tends to a nontrivial limiting curve,
given by a relatively simple equation.Comment: 39 pages, 2 figures; minor correction
Comments on the non-conformal gauge theories dual to Ypq manifolds
We study the infrared behavior of the entire class of Y(p,q) quiver gauge
theories. The dimer technology is exploited to discuss the duality cascades and
support the general belief about a runaway behavior for the whole family. We
argue that a baryonic classically flat direction is pushed to infinity by the
appearance of ADS-like terms in the effective superpotential. We also study in
some examples the IR regime for the L(a,b,c) class showing that the same
situation might be reproduced in this more general case as well.Comment: 48 pages, 27 figures; updated reference
Belief Propagation and Loop Series on Planar Graphs
We discuss a generic model of Bayesian inference with binary variables
defined on edges of a planar graph. The Loop Calculus approach of [1, 2] is
used to evaluate the resulting series expansion for the partition function. We
show that, for planar graphs, truncating the series at single-connected loops
reduces, via a map reminiscent of the Fisher transformation [3], to evaluating
the partition function of the dimer matching model on an auxiliary planar
graph. Thus, the truncated series can be easily re-summed, using the Pfaffian
formula of Kasteleyn [4]. This allows to identify a big class of
computationally tractable planar models reducible to a dimer model via the
Belief Propagation (gauge) transformation. The Pfaffian representation can also
be extended to the full Loop Series, in which case the expansion becomes a sum
of Pfaffian contributions, each associated with dimer matchings on an extension
to a subgraph of the original graph. Algorithmic consequences of the Pfaffian
representation, as well as relations to quantum and non-planar models, are
discussed.Comment: Accepted for publication in Journal of Statistical Mechanics: theory
and experimen