168 research outputs found

    Asymptotic Expansions for lambda_d of the Dimer and Monomer-Dimer Problems

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    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

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    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

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    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

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    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

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    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

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    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

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    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

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    We propose a new method for the problems of computing free energy and surface pressure for various statistical mechanics models on a lattice Zd\Z^d. Our method is based on representing the free energy and surface pressure in terms of certain marginal probabilities in a suitably modified sublattice of Zd\Z^d. 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 Z3\Z^3 belongs to the interval [0.78595,0.78599][0.78595,0.78599], improving best previously known estimate of (approximately) [0.7850,0.7862][0.7850,0.7862] obtained in \cite{FriedlandPeled},\cite{FriedlandKropLundowMarkstrom}. Moreover, we show that given a target additive error ϵ>0\epsilon>0, the computational effort of our method for these two models is (1/ϵ)O(1)(1/\epsilon)^{O(1)} \emph{both} for free energy and surface pressure. In contrast, prior methods, such as transfer matrix method, require exp((1/ϵ)O(1))\exp\big((1/\epsilon)^{O(1)}\big) computation effort.Comment: 33 pages, 4 figure

    The critical fugacity for surface adsorption of self-avoiding walks on the honeycomb lattice is 1+21+\sqrt{2}

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    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 μ=2+2.\mu=\sqrt{2+\sqrt{2}}. A key identity used in that proof was later generalised by Smirnov so as to apply to a general O(n) loop model with n[2,2]n\in [-2,2] (the case n=0n=0 corresponding to SAWs). We modify this model by restricting to a half-plane and introducing a surface fugacity yy 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 yc=1+2/2n.y_{\rm c}=1+2/\sqrt{2-n}. 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 n=0n=0, 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 TT, taken at its critical point 1/μ1/\mu, tends to 0 as TT 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 Zd\Z^d

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    In this short note we consider mixed short-long range independent bond percolation models on Zk+d\Z^{k+d}. Let puvp_{uv} be the probability that the edge (u,v)(u,v) will be open. Allowing a x,yx,y-dependent length scale and using a multi-scale analysis due to Aizenman and Newman, we show that the long distance behavior of the connectivity τxy\tau_{xy} is governed by the probability pxyp_{xy}. The result holds up to the critical point.Comment: 6 page
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