Einleitung

Fitzner, Werner

doi:10.14361/9783839435984-001

Cited by 2 publications

(11 citation statements)

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“…For the SAW, we also derived a NoBLE and implemented the bootstrap. In this way, we can show that mean-field behavior holds for SAW in d ≥ 7, see [15] and [14]. While the proof for d ≥ 7 is relatively simple, we expect that an extension of the technique to d = 5, 6 will not produce a substantially simpler proof than that of Hara and Slade, that is already optimal in the sense that it proves the result in all dimension above the SAW-upper critical dimension 4.…”

Section: Diagrammatic Boundsmentioning

confidence: 96%

“…In this paper and in a model-independent way, we perform the analysis in the third step and explain the numerical computations of the fourth step. The numerical computations and the explicit checks of the sufficient conditions for the NoBLE analysis to be successful are done in Mathematica notebooks that are available on the website of Robert Fitzner [14].…”

Section: Motivationmentioning

confidence: 99%

“…From private communication, we have learned that Takashi Hara also managed to prove that percolation in dimension d ≥ 15 obeys the infrared bound, although this result has not appeared in print. We hope that our method, as well as the accompanying Mathematica notebooks that are publicly available [14] to anyone who is interested, increase the transparency of the proof of the infrared bound for all the models involved.…”

Section: Diagrammatic Boundsmentioning

confidence: 99%

“…In high dimensions, short connections typically give the leading contribution to diagrams, so treating them more precisely often pays off. This requires the computation of a n (x), for all x ∈ Z d and n ∈ N. For n < 10, we compute these values using a simple Java program that can be downloaded from the website of the first author [14]. We start by explaining the bound on the repulsive bound for the example of a bubble.…”

Section: Bounds On Simple Diagramsmentioning

confidence: 99%

“…These numerical bounds are explained in detail in Section 5, using the ideas by Hara and Slade in [27]. These values are formulated in terms of integrals of Bessel-functions and are computed analytically, up to a specified precision in the Mathematica notebook available at [14]. This notebook needs to be compiled before the model-dependent parts can be performed, as the model-dependent analyses use them.…”

Section: Completion Of the Bootstrap Argument And Conclusionmentioning

confidence: 99%

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Generalized approach to the non-backtracking lace expansion

Fitzner

Hofstad

2016

Probab. Theory Relat. Fields

126

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The lace expansion is a powerful perturbative technique to analyze the critical behavior of random spatial processes such as the self-avoiding walk, percolation and lattice trees and animals. The non-backtracking lace expansion (NoBLE) is a modification that allows us to improve its applicability in the nearest-neighbor setting on the Z d -lattice for percolation, lattice trees and lattice animals.The NoBLE gives rise to a recursive formula that we study in this paper at a general level. We state assumptions that guarantee that the solution of this recursive formula satisfies the infrared bound. In two related papers, we show that these conditions are satisfied for percolation in d ≥ 11, for lattice trees in d ≥ 16 and for lattice animals in d ≥ 18.

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Section: Diagrammatic Boundsmentioning

confidence: 96%

Section: Motivationmentioning

confidence: 99%

Section: Diagrammatic Boundsmentioning

confidence: 99%

Section: Bounds On Simple Diagramsmentioning

confidence: 99%

Section: Completion Of the Bootstrap Argument And Conclusionmentioning

confidence: 99%

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Generalized approach to the non-backtracking lace expansion

Fitzner

Hofstad

2016

Probab. Theory Relat. Fields

126

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Mean-field behavior for nearest-neighbor percolation in $d>10$

Fitzner¹,

Hofstad²

2017

Electron. J. Probab.

100

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We prove that nearest-neighbor percolation in dimensions d ≥ 11 displays mean-field behavior by proving that the infrared bound holds, in turn implying the finiteness of the percolation triangle diagram. The finiteness of the triangle implies the existence and mean-field values of various critical exponents, such as γ = 1, β = 1, δ = 2. We also prove sharp x-space asymptotics for the two-point function and the existence of various arm exponents. Such results had previously been obtained in unpublished work by Hara and Slade for nearest-neighbor percolation in dimension d ≥ 19, so that we bring the dimension above which mean-field behavior is rigorously proved down from 19 to 11. Our results also imply sharp bounds on the critical value of nearest-neighbor percolation on Z d , which are provably at most 1.306% off in d = 11. We make use of the general method analyzed in [16], which proposes to use a lace expansion perturbing around non-backtracking random walk. This proof is computer-assisted, relying on (1) rigorous numerical upper bounds on various simple random walk integrals as proved by Hara and Slade [24]; and (2) a verification that the numerical conditions in [16] hold true. These two ingredients are implemented in two Mathematica notebooks that can be downloaded from the website of the first author.The main steps of this paper are (a) to derive a non-backtracking lace expansion for the percolation two-point function; (b) to bound the non-backtracking lace expansion coefficients, thus showing that the general methodology of [16] applies, and (c) to describe the numerical bounds on the coefficients.In the appendix of this extended version of the paper, we give additional details about the bounds on the NoBLE coefficients that are not given in the article version.

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Einleitung

Cited by 2 publications

References 0 publications

Generalized approach to the non-backtracking lace expansion

Generalized approach to the non-backtracking lace expansion

Mean-field behavior for nearest-neighbor percolation in $d>10$

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