On graph-like continua of finite length

Georgakopoulos, Agelos

doi:10.1016/j.topol.2014.05.017

Cited by 8 publications

(22 citation statements)

References 23 publications

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“…Our work in a sense partially generalizes the dendrite fractals considered in [Kig95], see also [Cro12] and references therein. Note also recent topological results on very similar spaces in [Geo14] as well as construction of Brownian motion on them in [GK]. In our work we attempt to appeal to two different communities that in present have small intersection: the fractal analysis community and the quantum graph community, hopefully generating a bidirectional flow of ideas from both fields.…”

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confidence: 99%

Energy and Laplacian on Hanoi-type fractal quantum graphs

Ruiz

Kelleher

Teplyaev

2016

J. Phys. A: Math. Theor.

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A. This article studies potential theory and spectral analysis on compact metric spaces, which we refer to as fractal quantum graphs. These spaces can be represented as a (possibly infinite) union of 1-dimensional intervals and a totally disconnected (possibly uncountable) compact set, which roughly speaking represents the set of junction points. Classical quantum graphs and fractal spaces such as the Hanoi attractor are included among them. We begin with proving the existence of a resistance form on the Hanoi attractor, and go on to establish heat kernel estimates and upper and lower bounds on the eigenvalue counting function of Laplacians corresponding to weakly self-similar measures on the Hanoi attractor. These estimates and bounds rely heavily on the relation between the length and volume scaling factors of the fractal. We then state and prove a necessary and sufficient condition for the existence of a resistance form on a general fractal quantum graph. Finally, we extend our spectral results to a large class of weakly self-similar fractal quantum graphs.C Date: September 6, 2018.

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confidence: 99%

Energy and Laplacian on Hanoi-type fractal quantum graphs

Ruiz

Kelleher

Teplyaev

2016

J. Phys. A: Math. Theor.

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“…Our bounds have interesting implications for Brownian motion on infinite networks as well as certain metric spaces. Motivated by a large body of literature on diffusions on fractals (see [21] or [17] and references therein), Georgakopoulos and Kolesko [18] have constructed a Brownian motion on a large class of metric spaces called graph-like spaces, which were introduced in [26] and can have a fractal structure [17]. Whenever such a space has finite 'length', that is, one-dimensional Hausdorff measure, our results imply that this Brownian motion will cover the whole space in finite expected time bounded by that length, thus almost surely in finite time.…”

Section: Our Resultsmentioning

confidence: 99%

New Bounds for Edge-Cover by Random Walk

Georgakopoulos¹,

Winkler²

2014

Combinator. Probab. Comp.

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We show that the expected time for a random walk on a (multi-)graph $G$ to traverse all $m$ edges of $G$, and return to its starting point, is at most $2m^2$; if each edge must be traversed in both directions, the bound is $3m^2$. Both bounds are tight and may be applied to graphs with arbitrary edge lengths, with implications for Brownian motion on a finite or infinite network of total edge-length $m$

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“…Similar objects have been recently investigated from a topological point of view in [15]; an stochastic approach of the construction of diffusion in that case has appeared in [16].…”

Section: Introductionmentioning

confidence: 99%