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Martinsson, A. (2015) *Accessibility percolation and first-passage percolation on the hypercube*. Göteborg : Chalmers University of Technology

** BibTeX **

@book{

Martinsson2015,

author={Martinsson, Anders},

title={Accessibility percolation and first-passage percolation on the hypercube},

abstract={<p>
In this thesis, we consider two percolation models on the n-dimensional binary hypercube, known as accessibility percolation and first-passage percolation. First-passage percolation randomly assigns non-negative weights, called passage times, to the edges of a graph and considers the minimal total weight of a path between given end-points. This quantity is called the first-passage time. Accessibility percolation is a biologically inspired model which has appeared in the mathematical literature only recently. Here, the vertices of a graph are randomly assigned heights, or fitnesses, and a path is considered accessible if strictly ascending. We let <img src="http://latex.codecogs.com/gif.latex?\mathbf{\hat{0}}" border="0"/> and <img src="http://latex.codecogs.com/gif.latex?\mathbf{\hat{1}}" border="0"/> denote the all zeroes and all ones vertices respectively.
</p><p>
A natural simplification of both models is the restriction to oriented paths, i.e. paths that only flip 0:s to 1:s. Paper I considers the existence of such accessible paths between <img src="http://latex.codecogs.com/gif.latex?\mathbf{\hat{0}}" border="0"/> and <img src="http://latex.codecogs.com/gif.latex?\mathbf{\hat{1}}" border="0"/> for fitnesses assigned according to the so-called House-of-Cards and Rough Mount Fuji models. In Paper II we consider the first-passage time between <img src="http://latex.codecogs.com/gif.latex?\mathbf{\hat{0}}" border="0"/> and <img src="http://latex.codecogs.com/gif.latex?\mathbf{\hat{1}}" border="0"/> in the case of independent standard exponential passage times. It is previously known that, in the oriented case, this quantity tends to 1 in probability as n tends to infinity. We show that without this restriction, the limit is instead <img src="http://latex.codecogs.com/gif.latex?\ln(1+\sqrt{2})\approx 0.88" border="0"/>. By adapting ideas in Paper II to unoriented accessibility percolation, we are able to determine a limiting probability for the existence of accessible paths from <img src="http://latex.codecogs.com/gif.latex?\mathbf{\hat{0}}" border="0"/> to the global fitness maximum. This is presented in Paper III.
</p>},

publisher={Institutionen för matematiska vetenskaper, matematik, Chalmers tekniska högskola,},

place={Göteborg},

year={2015},

keywords={hypercube, percolation, accessible path, house of cards, rough mount Fuji, first-passage percolation, Richardson's model, branching translation process},

note={126},

}

** RefWorks **

RT Dissertation/Thesis

SR Electronic

ID 211700

A1 Martinsson, Anders

T1 Accessibility percolation and first-passage percolation on the hypercube

YR 2015

AB <p>
In this thesis, we consider two percolation models on the n-dimensional binary hypercube, known as accessibility percolation and first-passage percolation. First-passage percolation randomly assigns non-negative weights, called passage times, to the edges of a graph and considers the minimal total weight of a path between given end-points. This quantity is called the first-passage time. Accessibility percolation is a biologically inspired model which has appeared in the mathematical literature only recently. Here, the vertices of a graph are randomly assigned heights, or fitnesses, and a path is considered accessible if strictly ascending. We let <img src="http://latex.codecogs.com/gif.latex?\mathbf{\hat{0}}" border="0"/> and <img src="http://latex.codecogs.com/gif.latex?\mathbf{\hat{1}}" border="0"/> denote the all zeroes and all ones vertices respectively.
</p><p>
A natural simplification of both models is the restriction to oriented paths, i.e. paths that only flip 0:s to 1:s. Paper I considers the existence of such accessible paths between <img src="http://latex.codecogs.com/gif.latex?\mathbf{\hat{0}}" border="0"/> and <img src="http://latex.codecogs.com/gif.latex?\mathbf{\hat{1}}" border="0"/> for fitnesses assigned according to the so-called House-of-Cards and Rough Mount Fuji models. In Paper II we consider the first-passage time between <img src="http://latex.codecogs.com/gif.latex?\mathbf{\hat{0}}" border="0"/> and <img src="http://latex.codecogs.com/gif.latex?\mathbf{\hat{1}}" border="0"/> in the case of independent standard exponential passage times. It is previously known that, in the oriented case, this quantity tends to 1 in probability as n tends to infinity. We show that without this restriction, the limit is instead <img src="http://latex.codecogs.com/gif.latex?\ln(1+\sqrt{2})\approx 0.88" border="0"/>. By adapting ideas in Paper II to unoriented accessibility percolation, we are able to determine a limiting probability for the existence of accessible paths from <img src="http://latex.codecogs.com/gif.latex?\mathbf{\hat{0}}" border="0"/> to the global fitness maximum. This is presented in Paper III.
</p>

PB Institutionen för matematiska vetenskaper, matematik, Chalmers tekniska högskola,

LA eng

LK http://publications.lib.chalmers.se/records/fulltext/211700/211700.pdf

OL 30