Lattice Green's functions in all dimensions

Guttmann, A J

doi:10.1088/1751-8113/43/30/305205

Cited by 92 publications

(193 citation statements)

References 36 publications

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“…(11) This integral is the Lattice Green function [14], whose value at z = 1 is related to the probability of a random walker to return to the origin.…”

Section: Methods and Resultsmentioning

confidence: 99%

Correspondence between spanning trees and the Ising model on a square lattice

Viswanathan¹

2017

Phys. Rev. E

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An important problem in statistical physics concerns the fascinating connections between partition functions of lattice models studied in equilibrium statistical mechanics on the one hand and graph theoretical enumeration problems on the other hand. We investigate the nature of the relationship between the number of spanning trees and the partition function of the Ising model on the square lattice. The spanning tree generating function T (z) gives the spanning tree constant when evaluated at z = 1, while giving the lattice green function when differentiated. It is known that for the infinite square lattice the partition function Z(K) of the Ising model evaluated at the critical temperature K = Kc is related to T (1). Here we show that this idea in fact generalizes to all real temperatures. We prove that (Z(K) sech 2K ) 2 = k exp T (k) , where k = 2 tanh(2K) sech(2K). The identical Mahler measure connects the two seemingly disparate quantities T (z) and Z(K). In turn, the Mahler measure is determined by the random walk structure function. Finally, we show that the the above correspondence does not generalize in a straightforward manner to non-planar lattices.

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“…(11) This integral is the Lattice Green function [14], whose value at z = 1 is related to the probability of a random walker to return to the origin.…”

Section: Methods and Resultsmentioning

confidence: 99%

Correspondence between spanning trees and the Ising model on a square lattice

Viswanathan¹

2017

Phys. Rev. E

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“…Lattice Green functions P (0; ξ) are well known for the most common d-dimensional lattices (see for instance [4,8]). The corresponding Green function in Eq.…”

Section: Number Of Distinct Sites Visitedmentioning

confidence: 99%

Arrival Statistics and Exploration Properties of Mortal Walkers

Yuste

Abad

Lindenberg

2014

First-Passage Phenomena and Their Applications

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We study some of the salient features of the arrival statistics and exploration properties of mortal random walkers, that is, walkers that may die as they move, or as they wait to move. Such evanescence or death events have profound consequences for quantities such as the number of distinct sites visited which are relevant for the computation of encountercontrolled rates in chemical kinetics. We exploit the observation that well-known methods developed decades ago for immortal walkers are widely applicable to mortal walkers. The particular cases of exponential and power-law evanescence are considered in detail. Finally, we discuss the relevance of our results to the target problem with mortal traps and a particular application thereof, namely, the defect diffusion model. Evanescence of defects is postulated as a possible complementary contribution or perhaps even an alternative to anomalous diffusion to explain observed stretched exponential relaxation behavior.

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“…Although the LDOS is sampled over a finite region, we stress that the host system is in the thermodynamic limit. The accurate calculation of lattice Green functions G 0 (r i ,r j ,ω) is itself a subtle and well-studied problem [46][47][48][49]. Exact diagonalization of finite-sized lattices or discrete Fourier transforms yield poor approximations to Green functions of the desired (semi-)infinite systems, especially at low scanning energies or near Van Hove singularities.…”

Section: A Real-space Formulationmentioning

confidence: 99%

Quasiparticle interference from magnetic impurities

2015

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Fourier transform scanning tunneling spectroscopy (FT-STS) measures the scattering of conduction electrons from impurities and defects, giving information about the electronic structure of both the host material and adsorbed impurities. We interpret such FT-STS measurements in terms of the quasiparticle interference (QPI), investigating the QPI due to magnetic impurities adsorbed on a range of representative non-magnetic host surfaces, and contrasting with the case of a simple scalar impurity or point defect. We demonstrate how the electronic correlations present for magnetic impurities markedly affect the QPI, showing e.g. a large intensity enhancement due to the Kondo effect, and universality at low temperatures/scanning-energies. The commonly-used joint density of states (JDOS) interpretation of FT-STS measurements is also considered, and shown to be insuffcient in many cases, including that of magnetic impurities.Comment: 18 pages, 12 figure

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Lattice Green's functions in all dimensions

Cited by 92 publications

References 36 publications

Correspondence between spanning trees and the Ising model on a square lattice

Correspondence between spanning trees and the Ising model on a square lattice

Arrival Statistics and Exploration Properties of Mortal Walkers

Quasiparticle interference from magnetic impurities

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