Critical exponents for a percolation model on transient graphs

Drewitz, Alexander; Prévost, Alexis; Rodriguez, Pierre-François

doi:10.1007/s00222-022-01168-z

Cited by 17 publications

(2 citation statements)

References 43 publications

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“…In the last decade, it has been subject to intensive research after a non‐trivial percolation phase transition has been identified for this model on

\mathbb {Z}^d

in [27], see for instance [2, 9, 12, 29]. Only very recently, in the remarkable paper [15], it has been shown that this phase transition is sharp, and, rather amazingly, the critical exponents have been identified for a related model of the level‐set percolation on the cable system of

\mathbb {Z}^d

in [13].…”

Section: Introductionmentioning

confidence: 99%

Giant component for the supercritical level‐set percolation of the Gaussian free field on regular expander graphs

Černý

2023

Comm Pure Appl Math

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We consider the zero‐average Gaussian free field on a certain class of finite d‐regular graphs for fixed . This class includes d‐regular expanders of large girth and typical realisations of random d‐regular graphs. We show that the level set of the zero‐average Gaussian free field above level h has a giant component in the whole supercritical phase, that is for all , with probability tending to one as the size of the graphs tends to infinity. In addition, we show that this component is unique. This significantly improves the result of [4], where it was shown that a linear fraction of vertices is in mesoscopic components if , and together with the description of the subcritical phase from [4] establishes a fully‐fledged percolation phase transition for the model.

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“…In the last decade, it has been subject to intensive research after a non‐trivial percolation phase transition has been identified for this model on

\mathbb {Z}^d

\mathbb {Z}^d

in [13].…”

Section: Introductionmentioning

confidence: 99%

Giant component for the supercritical level‐set percolation of the Gaussian free field on regular expander graphs

Černý

2023

Comm Pure Appl Math

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“…Consumers who purchase products publish their usage experiences on the www.ijacsa.thesai.org internet, and over time, their word-of-mouth (WoM) accumulates or decreases, leading to the diffusion of product information. This diffusion effect affects the number of consumers attracted during the normal sales phase [15][16]. The diffusion model of Base is shown in Fig.…”

mentioning

confidence: 99%

The Application of Cognitive Decision-Making Algorithm in Cross-Border e-Commerce Digital Marketing

Wang

2023

IJACSA

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Extensive global research aims to improve digital marketing profits through pricing decision-making and optimization. A dual word-of-mouth diffusion pricing model is developedfor cross-border e-commerce, addressing word-of-mouth accumulation and information diffusion effects. The traditional artificial bee colony algorithm is optimized with security domain search and information diffusion profiles, enhancing global search capabilities. Performance tests reveal that word-of-mouth scale significantly influences cross-border e-commerce profits, increasing with scale coefficient, consumer conversions, and optimal profits. The proposed algorithms demonstrate high efficiency and convergence rates, surpassing common iterations and benefits in the clothing pricing problem. The comprehensive imitation effect is -0.14, and the word-of-mouth scale effect is 1.34. Pre-sale and sales prices for clothing are set at 347.49 and 641.393, respectively. Similarly, in pricing cross-border e-commerce electronic products, the algorithm achieves optimal profits after 230 iterations, surpassing other algorithms. Overall, the proposed model exhibits superior computational performance in cross-border e-commerce pricing decision-making compared to conventional approaches.

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On the cost of the bubble set for random interlacements

Sznitman

2023

Invent. math.

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The main focus of this article concerns the strongly percolative regime of the vacant set of random interlacements on ${\mathbb{Z}}^{d}$ Z d , $d \ge 3$ d ≥ 3 . We investigate the occurrence in a large box of an excessive fraction of sites that get disconnected by the interlacements from the boundary of a concentric box of double size. The results significantly improve our findings in Sznitman (Probab. Math. Phys. 2–3:563–311, 2021). In particular, if, as expected, the critical levels for percolation and for strong percolation of the vacant set of random interlacements coincide, the asymptotic upper bound that we derive here matches in principal order a previously known lower bound. A challenging difficulty revolves around the possible occurrence of droplets that could get secluded by the random interlacements and thus contribute to the excess of disconnected sites, somewhat in the spirit of the Wulff droplets for Bernoulli percolation or for the Ising model. This feature is reflected in the present context by the so-called bubble set, a possibly quite irregular random set. A pivotal progress in this work has to do with the improved construction of a coarse grained random set accounting for the cost of the bubble set. This construction heavily draws both on the method of enlargement of obstacles originally developed in the mid-nineties in the context of Brownian motion in a Poissonian potential in Sznitman (Ann. Probab. 25(3):1180–1209, 1997; Brownian Motion, Obstacles and Random Media. Springer, Berlin, 1998), and on the resonance sets recently introduced by Nitzschner and Sznitman in (J. Eur. Math. Soc. 22(8):2629–2672, 2020) and further developed in a discrete set-up by Chiarini and Nitzschner in (Commun. Math. Phys. 386(3):1685–1745, 2021).

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Critical exponents for a percolation model on transient graphs

Cited by 17 publications

References 43 publications

Giant component for the supercritical level‐set percolation of the Gaussian free field on regular expander graphs

Giant component for the supercritical level‐set percolation of the Gaussian free field on regular expander graphs

The Application of Cognitive Decision-Making Algorithm in Cross-Border e-Commerce Digital Marketing

On the cost of the bubble set for random interlacements

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