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Anders Martinsson

Publications and source records attributed to Anders Martinsson.

43 records · Page 3Linked to original sources
arXiv2015

Permutations destroying arithmetic progressions in finite cyclic groups

A permutation πof an abelian group G is said to destroy arithmetic progressions (APs) if, whenever (a,b,c) is a non-trivial 3-term AP in G, that is c-b=b-a and a,b,c are not all equal, then (π(a),π(b),π(c)) is not an AP. In a paper from 2004, the first author conjectured that such a permutation exists of Z/nZ, for all n except 2,3,5 and 7. Here we prove, as a special case of a more general result, that such a permutation exists for all n >= n_0, for some explcitly constructed number n_0 \approx 1.4 x 10^{14}. We also construct such a permutation of Z/pZ for all primes p > 3 such that p = 3 (mod 8).

math.NT↗
arXiv2015

The Hegselmann-Krause dynamics on the circle converge

We consider the Hegselmann-Krause dynamics on a one-dimensional torus and provide the first proof of convergence of this system. The proof requires only fairly minor modifications of existing methods for proving convergence in Euclidean space.

eess.SY↗
arXiv2015

Accessibility percolation and first-passage site percolation on the unoriented binary hypercube

Inspired by biological evolution, we consider the following so-called accessibility percolation problem: The vertices of the unoriented $n$-dimensional binary hypercube are assigned independent $U(0, 1)$ weights, referred to as fitnesses. A path is considered accessible if fitnesses are strictly increasing along it. We prove that the probability that the global fitness maximum is accessible from the all zeroes vertex converges to $1-\frac{1}{2}\ln\left(2+\sqrt{5}\right)$ as $n\rightarrow\infty$. Moreover, we prove that if one conditions on the location of the fitness maximum being $v$, then provided $v$ is not too close to the all zeroes vertex in Hamming distance, the probability that $v$ is accessible converges to a function of this distance divided by $n$ as $n\rightarrow\infty$. This resolves a conjecture by Berestycki, Brunet and Shi in almost full generality. As a second result we show that, for any graph, accessibility percolation can equivalently be formulated in terms of first-passage site percolation. This connection is of particular importance for the study of accessibility percolation on trees.

math.PR↗
arXiv2014

Unoriented first-passage percolation on the n-cube

The $n$-dimensional binary hypercube is the graph whose vertices are the binary $n$-tuples $\{0, 1\}^n$ and where two vertices are connected by an edge if they differ at exactly one coordinate. We prove that if the edges are assigned independent mean 1 exponential costs, the minimum length $T_n$ of a path from $(0, 0, \dots, 0)$ to $(1, 1, \dots, 1)$ converges in probability to $\ln(1+\sqrt{2}) \approx 0.881$. It has previously been shown by Fill and Pemantle (1993) that this so-called first-passage time asymptotically almost surely satisfies $\ln(1+\sqrt{2}) - o(1) \leq T_n \leq 1+o(1)$, and has been conjectured to converge in probability by Bollobás and Kohayakawa (1997). A key idea of our proof is to consider a lower bound on Richardson's model, closely related to the branching process used in the article by Fill and Pemantle to obtain the bound $T_n \geq \ln\left(1+\sqrt{2}\right)-o(1)$. We derive an explicit lower bound on the probability that a vertex is infected at a given time. This result is formulated for a general graph and may be applicable in a more general setting.

math.PR↗
arXiv2014

On the existence of accessible paths in various models of fitness landscapes

We present rigorous mathematical analyses of a number of well-known mathematical models for genetic mutations. In these models, the genome is represented by a vertex of the $n$-dimensional binary hypercube, for some $n$, a mutation involves the flipping of a single bit, and each vertex is assigned a real number, called its fitness, according to some rules. Our main concern is with the issue of existence of (selectively) accessible paths; that is, monotonic paths in the hypercube along which fitness is always increasing. Our main results resolve open questions about three such models, which in the biophysics literature are known as house of cards (HoC), constrained house of cards (CHoC) and rough Mount Fuji (RMF). We prove that the probability of there being at least one accessible path from the all-zeroes node $\mathbf {v}^0$ to the all-ones node $\mathbf {v}^1$ tends respectively to 0, 1 and 1, as $n$ tends to infinity. A crucial idea is the introduction of a generalization of the CHoC model, in which the fitness of $\mathbf {v}^0$ is set to some $α=α_n\in[0,1]$. We prove that there is a very sharp threshold at $α_n=\frac{\ln n}{n}$ for the existence of accessible paths from $\mathbf {v}^0$ to $\mathbf {v}^1$. As a corollary we prove significant concentration, for $α$ below the threshold, of the number of accessible paths about the expected value (the precise statement is technical; see Corollary 1.4). In the case of RMF, we prove that the probability of accessible paths from $\mathbf {v}^0$ to $\mathbf {v}^1$ existing tends to $1$ provided the drift parameter $θ=θ_n$ satisfies $nθ_n\rightarrow\infty$, and for any fitness distribution which is continuous on its support and whose support is connected.

math.PR↗
arXiv2013

A variant of the multi-agent rendezvous problem

The classical multi-agent rendezvous problem asks for a deterministic algorithm by which $n$ points scattered in a plane can move about at constant speed and merge at a single point, assuming each point can use only the locations of the others it sees when making decisions and that the visibility graph as a whole is connected. In time complexity analyses of such algorithms, only the number of rounds of computation required are usually considered, not the amount of computation done per round. In this paper, we consider $Ω(n^2 \log n)$ points distributed independently and uniformly at random in a disc of radius $n$ and, assuming each point can not only see but also, in principle, communicate with others within unit distance, seek a randomised merging algorithm which asymptotically almost surely (a.a.s.) runs in time O(n), in other words in time linear in the radius of the disc rather than in the number of points. Under a precise set of assumptions concerning the communication capabilities of neighboring points, we describe an algorithm which a.a.s. runs in time O(n) provided the number of points is $o(n^3)$. Several questions are posed for future work.

cs.MA↗