Computability of series representations for Green's functions in a rectangle

Melnikov, Yuri A.; Melnikov, Max Y.

doi:10.1016/j.enganabound.2006.03.010

Cited by 11 publications

(8 citation statements)

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“…We may note that ( 11)-( 14) are associated with the Chebyshev polynomials of the second kind U n (x) = sin(n + 1)θ/ sin θ, cos θ = x, which can be shown by using the formula (62). We can also check that the partition functions ( 11)-( 14) are identical to (7), ( 8) and ( 1), with the aid of the identities ( 64) and (65).…”

Section: Grassmannian Representation Of the Dimer Modelmentioning

confidence: 99%

The Expectation Value of the Number of Loops and the Left-Passage Probability in the Double-Dimer Model

Ghodratipour

Rouhani

2019

Commun. Math. Phys.

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We study various statistical properties of the double-dimer model, a generalization of the dimer model, on rectangular domains of the square lattice. We take advantage of the Grassmannian representation of the dimer model, first to calculate the probability distribution of nontrivial loops around a cylinder, which is consistent with the previously known result, and then to calculate the expectation value of the number of loops surrounding two faces and the left-passage probability, both in the discrete and the continuum cases. We also briefly explain the calculation of some related observables. As a by-product, we obtain the partition function of the dimer model in the presence of two and four monomers, and a single monomer on the boundary. * ghodratipour n@physics.sharif.edu † srouhani@sharif.ir 1 arXiv:1805.03930v2 [cond-mat.stat-mech] 27 Jun 2018 statistical physics, including the O(n) models. These are a large class of critical systems in physics, conjectured to be related to SLE κ and CLE κ concisely expressed by the formula n = −2 cos(4π/κ). The parameter κ is so in significant relation to the central charge c in CFT and as a fact, SLEs (CLEs) with κ ≤ 4 are believed to be the minimal models of CFT with c in [0, 1] [21-23]. One prototypical model where such issues are concerned is the two-dimensional Gaussian Free Field (GFF).The two-dimensional GFF is a central object both in physics and mathematics, related to the Coulomb gas, which falls into the same universality class as other statistical mechanical models such as the XY-model and the two-dimensional sine-Gordon model [17,24,25]. It is the mathematical model of two-dimensional bosonic CFT, and is a two-dimensional analog of the Brownian motion [26]. The notions associated with the Brownian motion, such as the Gaussian measure, the Laplacian, the Green's function and consequently conformal invariance, are also substantial in GFF [25,27], and many stochastic differential equations in two dimensions depend on it as those in one dimension on the Brownian motion [28,29]. We can practically think of GFF as a natural model of random height functions though it is actually a generalized function or a distribution (in sense of Schwartz [26]). It yields a dual interpretation of many loop models, especially the O(n) models, which in SLE/CLE formalism, is very natural in the case of κ = 4 (equivalently n = 2); the zero level line of the Gaussian free field (with appropriate boundary conditions) is a chordal SLE 4 and the collection of level loops of it corresponds to CLE 4 [30,31].Besides, there is a family of conformally invariant loop models in two dimensions, the Brownian Loop Soups (BLSs), which is deeply related to SLE and CLE [20,32]. Each BLS is a Poisson point process of Brownian loops [33], with intensity parameter related to κ via the equation c = (3κ − 8)(6 − κ)/2κ [32], indicating that for low enough intensities the outer boundaries of clusters of Brownian loops are distributed like CLEs for the spectral parameter between 8/3 and 4. This implies that th...

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Section: Grassmannian Representation Of the Dimer Modelmentioning

confidence: 99%

The Expectation Value of the Number of Loops and the Left-Passage Probability in the Double-Dimer Model

Ghodratipour

Rouhani

2019

Commun. Math. Phys.

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“…Taking advantage of the defining properties of  ( ) , G P Q , it sounds reasonable to express the solution UQ (P) to the problem in ( 5) - (7) in the integral form…”

Section: Description Of the Gf-mfe Proceduresmentioning

confidence: 99%

“…Moreover, the boundary condition in ( 6) is also satisfied, regardless of the density functions μj(Q). Thus, the presence of the latter provides the representation in (8) with some degree of freedom, which can be used to satisfy the boundary conditions in (7). Taking the field point P in (8) to the factual aperture contours Γ j ; (j=1; m), we obtain an m×m system of regular functional equations in the density functions μ j (S).…”

Section: Description Of the Gf-mfe Proceduresmentioning

confidence: 99%

“…The convergence rate of the series in (45) is low (1/n) for the diagonal elements. To enhance it, one can use the procedure proposed in [6] and [7]. This reduces the element to the computer-friendly from…”

Section: Resolving Green's Functionsmentioning

confidence: 99%

“…The opinion is senseless, but its very existence is hard unfortunately to ignore. Our long time [4,[5][6][7][8] involvement with the boundary integral equation method reveals high computational potential of a specific modification of the method of functional equations, which we originally proposed in [4]. The modification utilizes some Green's functions relevant to the considered problem, and will be called herein the Green's function modification of the method of functional equations (abbreviated as GF-MFE).…”

Section: Introductionmentioning

confidence: 99%

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To the Efficiency of a Green's Function Modification of the Method of Functional Equations

Melnikov¹

2014

J Appl Computat Math

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Analysis of the computational potential is provided for a modification of one of the numerical approaches to the classical boundary integral equations method. The originally proposed name of the approach is the method of functional equations, but in nowadays it is also referred to as the fundamental solutions method. Undesired after effects of this name flip are pointed out. The modification, implemented in this study, requires computer-friendly representations for some relevant Green's function and significantly enhances the resolving potential of the method. This work focuses on the exploration of the computational applicability of this modification. Chosen for that boundaryvalue problems and stated on regions of irregular configuration for second order elliptic equations with discontinuous coefficients.

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Introduction

Melnikov¹

2011

Green's Functions and Infinite Products

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Computability of series representations for Green's functions in a rectangle

Cited by 11 publications

References 1 publication

The Expectation Value of the Number of Loops and the Left-Passage Probability in the Double-Dimer Model

The Expectation Value of the Number of Loops and the Left-Passage Probability in the Double-Dimer Model

To the Efficiency of a Green's Function Modification of the Method of Functional Equations

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