168 research outputs found
Asymptotic Expansions for lambda_d of the Dimer and Monomer-Dimer Problems
In the past few years we have derived asymptotic expansions for lambda_d of
the dimer problem and lambda_d(p) of the monomer-dimer problem. The many
expansions so far computed are collected herein. We shine a light on results in
two dimensions inspired by the work of M. E. Fisher. Much of the work reported
here was joint with Shmuel Friedland.Comment: 4 page
An Asymptotic Expansion and Recursive Inequalities for the Monomer-Dimer Problem
Let (lambda_d)(p) be the p monomer-dimer entropy on the d-dimensional integer
lattice Z^d, where p in [0,1] is the dimer density. We give upper and lower
bounds for (lambda_d)(p) in terms of expressions involving (lambda_(d-1))(q).
The upper bound is based on a conjecture claiming that the p monomer-dimer
entropy of an infinite subset of Z^d is bounded above by (lambda_d)(p). We
compute the first three terms in the formal asymptotic expansion of
(lambda_d)(p) in powers of 1/d. We prove that the lower asymptotic matching
conjecture is satisfied for (lambda_d)(p).Comment: 15 pages, much more about d=1,2,
Weighted distances in scale-free preferential attachment models
We study three preferential attachment models where the parameters are such
that the asymptotic degree distribution has infinite variance. Every edge is
equipped with a non-negative i.i.d. weight. We study the weighted distance
between two vertices chosen uniformly at random, the typical weighted distance,
and the number of edges on this path, the typical hopcount. We prove that there
are precisely two universality classes of weight distributions, called the
explosive and conservative class. In the explosive class, we show that the
typical weighted distance converges in distribution to the sum of two i.i.d.
finite random variables. In the conservative class, we prove that the typical
weighted distance tends to infinity, and we give an explicit expression for the
main growth term, as well as for the hopcount. Under a mild assumption on the
weight distribution the fluctuations around the main term are tight.Comment: Revised version, results are unchanged. 30 pages, 1 figure. To appear
in Random Structures and Algorithm
Localization criteria for Anderson models on locally finite graphs
We prove spectral and dynamical localization for Anderson models on locally
finite graphs using the fractional moment method. Our theorems extend earlier
results on localization for the Anderson model on \ZZ^d. We establish
geometric assumptions for the underlying graph such that localization can be
proven in the case of sufficiently large disorder
Extracting the bulk metric from boundary information in asymptotically AdS spacetimes
We use geodesic probes to recover the entire bulk metric in certain
asymptotically AdS spacetimes. Given a spectrum of null geodesic endpoints on
the boundary, we describe two remarkably simple methods for recovering the bulk
information. After examining the issues which affect their application in
practice, we highlight a significant advantage one has over the other from a
computational point of view, and give some illustrative examples. We go on to
consider spacetimes where the methods cannot be used to recover the complete
bulk metric, and demonstrate how much information can be recovered in these
cases.Comment: 33 pages, 11 figures; v2 references adde
Thermodynamics of Large AdS Black Holes
We consider leading order quantum corrections to the geometry of large AdS
black holes in a spherical reduction of four-dimensional Einstein gravity with
negative cosmological constant. The Hawking temperature grows without bound
with increasing black hole mass, yet the semiclassical back-reaction on the
geometry is relatively mild, indicating that observers in free fall outside a
large AdS black hole never see thermal radiation at the Hawking temperature.
The positive specific heat of large AdS black holes is a statement about the
dual gauge theory rather than an observable property on the gravity side.
Implications for string thermodynamics with an AdS infrared regulator are
briefly discussed.Comment: 17 pages, 1 figure, v2. added reference
Optimal Design and Tolerancing of Compressor Blades Subject to Manufacturing Variability
This paper presents a computational approach for optimal robust design and tolerancing of turbomachinery compressor blades that are subject to geometric variability. This approach simultaneously determines the optimal blade geometry and manufacturing tolerances to minimize the overall cost of producing and operating the resulting compressor blades. A pathwise sensitivity method is used to compute gradient information that is in turn used to optimize the design and tolerances. Results for a two-dimensional subsonic compressor are presented, demonstrating the significant performance improvements that can be achieved using the proposed approach.Pratt & Whitney Aircraft CompanyBoeing Compan
Sequential cavity method for computing free energy and surface pressure
We propose a new method for the problems of computing free energy and surface
pressure for various statistical mechanics models on a lattice . Our
method is based on representing the free energy and surface pressure in terms
of certain marginal probabilities in a suitably modified sublattice of .
Then recent deterministic algorithms for computing marginal probabilities are
used to obtain numerical estimates of the quantities of interest. The method
works under the assumption of Strong Spatial Mixing (SSP), which is a form of a
correlation decay.
We illustrate our method for the hard-core and monomer-dimer models, and
improve several earlier estimates. For example we show that the exponent of the
monomer-dimer coverings of belongs to the interval ,
improving best previously known estimate of (approximately)
obtained in \cite{FriedlandPeled},\cite{FriedlandKropLundowMarkstrom}.
Moreover, we show that given a target additive error , the
computational effort of our method for these two models is
\emph{both} for free energy and surface pressure. In
contrast, prior methods, such as transfer matrix method, require
computation effort.Comment: 33 pages, 4 figure
The critical fugacity for surface adsorption of self-avoiding walks on the honeycomb lattice is
In 2010, Duminil-Copin and Smirnov proved a long-standing conjecture of
Nienhuis, made in 1982, that the growth constant of self-avoiding walks on the
hexagonal (a.k.a. honeycomb) lattice is A key identity
used in that proof was later generalised by Smirnov so as to apply to a general
O(n) loop model with (the case corresponding to SAWs).
We modify this model by restricting to a half-plane and introducing a surface
fugacity associated with boundary sites (also called surface sites), and
obtain a generalisation of Smirnov's identity. The critical value of the
surface fugacity was conjectured by Batchelor and Yung in 1995 to be This value plays a crucial role in our generalized
identity, just as the value of growth constant did in Smirnov's identity.
For the case , corresponding to \saws\ interacting with a surface, we
prove the conjectured value of the critical surface fugacity. A crucial part of
the proof involves demonstrating that the generating function of self-avoiding
bridges of height , taken at its critical point , tends to 0 as
increases, as predicted from SLE theory.Comment: Major revision, references updated, 25 pages, 13 figure
Decay Properties of the Connectivity for Mixed Long Range Percolation Models on
In this short note we consider mixed short-long range independent bond
percolation models on . Let be the probability that the edge
will be open. Allowing a -dependent length scale and using a
multi-scale analysis due to Aizenman and Newman, we show that the long distance
behavior of the connectivity is governed by the probability
. The result holds up to the critical point.Comment: 6 page
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