arXiv subjects

Sven Erick Alm

Publications and source records attributed to Sven Erick Alm.

4 recordsLinked to original sources

arXiv2014

First passage percolation on $\mathbb{Z}^2$ -- a simulation study

First passage percolation on $\mathbb{Z}^2$ is a model for describing the spread of an infection on the sites of the square lattice. The infection is spread via nearest neighbor sites and the time dynamic is specified by random passage times attached to the edges. In this paper, the speed of the growth and the shape of the infected set is studied by aid of large-scale computer simulations, with focus on continuous passage time distributions. It is found that the most important quantity for determining the value of the time constant, which indicates the inverse asymptotic speed of the growth, is $\mathbf{E}[\min\{τ_1,\ldots,τ_4\}]$, where $τ_1,\ldots,τ_4$ are i.i.d. passage time variables. The relation is linear for a large class of passage time distributions. Furthermore, the directional time constants are seen to be increasing when moving from the axis towards the diagonal, so that the limiting shape is contained in a circle with radius defined by the speed along the axes. The shape comes closer to the circle for distributions with larger variability.

math.PR↗

arXiv2013

First critical probability for a problem on random orientations in $G(n,p)$

We study the random graph $G(n,p)$ with a random orientation. For three fixed vertices $s,a,b$ in $G(n,p)$ we study the correlation of the events $a \to s$ and $s\to b$. We prove that asymptotically the correlation is negative for small $p$, $p<\frac{C_1}n$, where $C_1\approx0.3617$, positive for $\frac{C_1}n<p<\frac2n$ and up to $p=p_2(n)$. Computer aided computations suggest that $p_2(n)=\frac{C_2}n$, with $C_2\approx7.5$. We conjecture that the correlation then stays negative for $p$ up to the previously known zero at $\frac12$; for larger $p$ it is positive.

math.PR↗

arXiv2010

Correlations for paths in random orientations of G(n,p) and G(n,m)

We study random graphs, both $G(n,p)$ and $G(n,m)$, with random orientations on the edges. For three fixed distinct vertices s,a,b we study the correlation, in the combined probability space, of the events a -> s and s -> b. For G(n,p), we prove that there is a p_c=1/2 such that for a fixed p p_c the correlation is positive for large enough n. We conjecture that for a fixed n\ge 27 the correlation changes sign three times for three critical values of p. For G(n,m) it is similarly proved that, with $p=m/\binom{n}{2}$, there is a critical p_c that is the solution to a certain equation and approximately equal to 0.7993. A lemma, which computes the probability of non existence of any k directed edges in G(n,m), is thought to be of independent interest. We present exact recursions to compute P(a -> s)$ and P(a -> s, s -> b)$. We also briefly discuss the corresponding question in the quenched version of the problem.

math.PR↗

arXiv2009

A counter-intuitive correlation in a random tournament

Consider a randomly oriented graph $G=(V,E)$ and let $a$, $s$ and $b$ be three distinct vertices in $V$. We study the correlation between the events $\{a\to s\}$ and $\{s\to b\}$. We show that, when $G$ is the complete graph $K_n$, the correlation is negative for $n=3$, zero for $n=4$, and that, counter-intuitively, it is positive for $n\ge 5$. We also show that the correlation is always negative when $G$ is a cycle, $C_n$, and negative or zero when $G$ is a tree (or a forest).

math.PR↗