arXiv subjects

Robert Fitzner

Publications and source records attributed to Robert Fitzner.

5 recordsLinked to original sources

arXiv2019

NoBLE for lattice trees and lattice animals

We study lattice trees (LTs) and animals (LAs) on the nearest-neighbor lattice $\mathbb{Z}^d$ in high dimensions. We prove that LTs and LAs display mean-field behavior above dimension $16$ and $17$, respectively. Such results have previously been obtained by Hara and Slade in sufficiently high dimensions. The dimension above which their results apply was not yet specified. We rely on the non-backtracking lace expansion (NoBLE) method that we have recently developed. The NoBLE makes use of an alternative lace expansion for LAs and LTs that perturbs around non-backtracking random walk rather than simple random walk, leading to smaller corrections. The NoBLE method then provides a careful computational analysis that improves the dimension above which the result applies. Universality arguments predict that the upper critical dimension, above which our results apply, is equal to $d_c=8$ for both models, as is known for sufficiently spread-out models by the results of Hara and Slade mentioned earlier. The main ingredients in this paper are (a) to derive a non-backtracking lace expansion for the LT and LA two-point functions; (b) to bound the non-backtracking lace-expansion coefficients, thus showing that our general NoBLE methodology can be applied; and (c) to obtain sharp numerical bounds on the coefficients. Our proof is complemented by a computer-assisted numerical analysis that verifies that the necessary bounds used in the NoBLE are satisfied.

math-ph↗

arXiv2016

Birds of a feather or opposites attract - effects in network modelling

We study properties of some standard network models when the population is split into two types and the connection pattern between the types is varied. The studied models are generalizations of the Erd\H{o}s-R\'{e}nyi graph, the configuration model and a preferential attachment graph. For the Erd\H{o}s-R\'{e}nyi graph and the configuration model, the focus is on the component structure. We derive expressions for the critical parameter, indicating when there is a giant component in the graph, and study the size of the largest component by aid of simulations. When the expected degrees in the graph are fixed and the connections are shifted so that more edges connect vertices of different types, we find that the critical parameter decreases. The size of the largest component in the supercritical regime can be both increasing and decreasing as the connections change, depending on the combination of types. For the preferential attachment model, we analyze the degree distributions of the two types and derive explicit expressions for the degree exponents. The exponents are confirmed by simulations that also illustrate other properties of the degree structure.

math.PR↗

arXiv2015

Generalized approach to the non-backtracking lace expansion

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 $\mathbb Z^d$-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 $d\geq 11$, for lattice trees in $d\geq 16$ and for lattice animals in $d\geq 21$.

math.PR↗

arXiv2015

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

We prove that nearest-neighbor percolation in dimensions $d\geq 11$ 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 $\gamma=1, \beta=1, \delta=2$. 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 $d\geq 19$, 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 $\mathbb{Z}^d$, which are provably at most $1.306\%$ off in $d=11$. We make use of the general method analyzed in the accompanying paper "Generalized approach to the non-backtracking lace expansion" by Fitzner and van der Hofstad, which proposes to use a lace expansion perturbing around non-backtracking random walk. This proof is {\em computer-assisted}, relying on (1) rigorous numerical upper bounds on various simple random walk integrals as proved by Hara and Slade (1992) and (2) a verification that the derived numerical conditions 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 applies, and (c) to describe the numerical bounds on the coefficients. In the appendix of this extended version, we give additional details about the bounds that are not given in the article version.

math.PR↗

arXiv2012

Non-backtracking random walk

We consider non-backtracking random walk (NBW) in the nearest-neighbor setting on the Zd-lattice and on tori. We evaluate the eigensystem of the m X m-dimensional transition matrix of NBW where m denote the degree of the graph. We use its eigensystem to show a functional central limit theorem for NBW on Zd and to obtain estimates on the convergence towards the stationary distribution for NBW on the torus.

math.PR↗