Generalized approach to the non-backtracking lace expansion

Fitzner, RJ Robert; Hofstad, RW Remco van der

doi:10.1007/s00440-016-0747-8

Cited by 15 publications

(130 citation statements)

References 39 publications

Supporting

Mentioning

126

Contrasting

Order By: Relevance

“…(II) Two other Mathematica notebooks, a first that implements the computations in our general approach to the non-backtracking lace expansion (NoBLE) in [16], as well as a notebook that computes the rigorous bounds on the lace-expansion coefficients provided in the present paper. These notebooks do nothing else than implement the bounds proved here and in [16], and rely on nothing but many multiplications, additions as well as diagonalizations of two three-by-three matrices. These computations could be performed by hand, but the use of the notebooks tremendously simplifies them.…”

Section: Motivationmentioning

confidence: 99%

“…(c) the analysis presented in [16] to obtain the infrared bound in Theorem 1.1 for all p ≤ p c for all d ≥ 11, as stated in Proposition 2.4; and (d) a computer-assisted proof to verify the numerical conditions arising in the analysis in [16].…”

Section: Philosophy Of the Proofmentioning

confidence: 99%

“…In Section 2.4, we explain how we obtain part (c) using the analysis of [16], the computer-assisted proof [13] of part (d) and the results of this paper. For the analysis in the generalized setting [16] we state assumptions, which we verify for percolation in Sections 2.4, 3.5 and 5.4, respectively. Part (d) is explained in detail in Section 2.5, where we describe how the necessary computations are performed in several Mathematica notebooks.…”

Section: Philosophy Of the Proofmentioning

confidence: 99%

“…Part (d) is explained in detail in Section 2.5, where we describe how the necessary computations are performed in several Mathematica notebooks. The mathematics behind the notebooks is explained in [16]. The notebooks also include a routine that verifies whether the numerical assumption on the expansions are satisfied and thereby verifies whether the analysis of [16] yields the infrared bounds for percolation or not for a given dimension.…”

Section: Philosophy Of the Proofmentioning

confidence: 99%

See 3 more Smart Citations

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

Fitzner¹,

Hofstad²

2017

Electron. J. Probab.

Self Cite

100

View full text Add to dashboard Cite

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.

show abstract

Section: Motivationmentioning

confidence: 99%

Section: Philosophy Of the Proofmentioning

confidence: 99%

Section: Philosophy Of the Proofmentioning

confidence: 99%

Section: Philosophy Of the Proofmentioning

confidence: 99%

See 2 more Smart Citations

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

Fitzner¹,

Hofstad²

2017

Electron. J. Probab.

Self Cite

100

View full text Add to dashboard Cite

show abstract

“…The study of critical percolation in high dimensions (currently meaning d 11; see [32,33]) saw significant progress through the use of techniques known as lace expansion (for a recent survey, see [41]). Those techniques allowed a deep understanding of critical clusters in high dimensions (see [36,38]) and in particular opened the door to the proof of the Alexander-Orbach conjecture mentioned above.…”

Section: Introductionmentioning

confidence: 99%

Scaling Limit for the Ant in High‐Dimensional Labyrinths

Arous

Cabezas

Fribergh

2019

Comm Pure Appl Math

View full text Add to dashboard Cite

We study here a detailed conjecture regarding one of the most important cases of anomalous diffusion, i.e., the behavior of the “ant in the labyrinth.” It is natural to conjecture that the scaling limit for random walks on large critical random graphs exists in high dimensions and is universal. This scaling limit is simply the natural Brownian motion on the integrated super‐Brownian excursion. We give here a set of four natural, sufficient conditions on the critical graphs and prove that this set of assumptions ensures the validity of this conjecture. The remaining future task is to prove that these sufficient conditions hold for the various classical cases of critical random structures, like the usual Bernoulli bond percolation, oriented percolation, and spread‐out percolation in high enough dimension. In a companion paper, we do precisely that in a first case, the random walk on the trace of a large critical branching random walk. We verify the validity of these sufficient conditions and thus obtain the scaling limit mentioned above in dimensions larger than 14. © 2019 Wiley Periodicals, Inc.

show abstract

Restricted Percolation Critical Exponents in High Dimensions

Chatterjee

Hanson

2020

Comm Pure Appl Math

View full text Add to dashboard Cite

Despite great progress in the study of critical percolation on ℤd for d large, properties of critical clusters in high‐dimensional fractional spaces and boxes remain poorly understood, unlike the situation in two dimensions. Closely related models such as critical branching random walk give natural conjectures for the value of the relevant high‐dimensional critical exponents; see in particular the conjecture by Kozma‐Nachmias that the probability that 0 and (n, n, n, …) are connected within [−n, n]d scales as n−2 − 2d. In this paper, we study the properties of critical clusters in high‐dimensional half‐spaces and boxes. In half‐spaces, we show that the probability of an open connection (“arm”) from 0 to the boundary of a sidelength n box scales as n−3. We also find the scaling of the half‐space two‐point function (the probability of an open connection between two vertices) and the tail of the cluster size distribution. In boxes, we obtain the scaling of the two‐point function between vertices which are any macroscopic distance away from the boundary. Our argument involves a new application of the “mass transport" principle which we expect will be useful to obtain quantitative estimates for a range of other problems. © 2020 Wiley Periodicals LLC

show abstract

Generalized approach to the non-backtracking lace expansion

Cited by 15 publications

References 39 publications

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

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

Scaling Limit for the Ant in High‐Dimensional Labyrinths

Restricted Percolation Critical Exponents in High Dimensions

Contact Info

Product

Resources

About