Related papers: Positive speed for high-degree automaton groups
We explore the use of machine-learning techniques to detect quantum speedup in random walks on graphs. Specifically, we investigate the performance of three different neural-network architectures (variations on fully connected and…
A self-avoiding walk (SAW) is a path on a graph that visits each vertex at most once. The mean square displacement of an $n$-step SAW is the expected value of the square of the distance between the ending point and the starting point of an…
A graph is said to be symmetric if its automorphism group is transitive on its arcs. Guo et al. (Electronic J. Combin. 18, \#P233, 2011) and Pan et al. (Electronic J. Combin. 20, \#P36, 2013) determined all pentavalent symmetric graphs of…
We consider a one-dimensional transient cookie random walk. It is known from a previous paper that a cookie random walk $(X_n)$ has positive or zero speed according to some positive parameter $\alpha >1$ or $\le 1$. In this article, we give…
We prove that if the group generated by a Mealy automaton acts level-transitively on a regular rooted tree, then the semigroup generated by the dual automaton has exponential growth, hence giving a decision procedure of exponential growth…
We extend the use of random evolving sets to time-varying conductance models and utilize it to provide tight heat kernel upper bounds. It yields the transience of any uniformly lazy random walk, on Z^d, d>=3, equipped with uniformly bounded…
We give faster algorithms for producing sparse approximations of the transition matrices of $k$-step random walks on undirected, weighted graphs. These transition matrices also form graphs, and arise as intermediate objects in a variety of…
Computing the rate of evolution in spatially structured populations is difficult. A key quantity is the fixation time of a single mutant with relative reproduction rate $r$ which invades a population of residents. We say that the fixation…
We prove that the speed of a biased random walk on a supercritical Galton-Watson tree conditioned to survive is analytic within the ballistic regime. This extends the previous work arXiv:1906.07913 in which it was shown that the speed is…
Aaronson and Ambainis (2009) and Chailloux (2018) showed that fully symmetric (partial) functions do not admit exponential quantum query speedups. This raises a natural question: how symmetric must a function be before it cannot exhibit a…
We study fine structure related to finitely supported random walks on infinite finitely generated discrete groups, largely motivated by dimension group techniques. The unfaithful extreme harmonic functions (defined only on proper space-time…
In this paper we consider a class of one-dimensional interacting particle systems in equilibrium, constituting a dynamic random environment, together with a nearest-neighbor random walk that on occupied/vacant sites has a local drift to the…
We develop a new framework for analysing finite connected, oriented graphs of valency 4, which admit a vertex-transitive and edge-transitive group of automorphisms preserving the edge orientation. We identify a sub-family of "basic" graphs…
A random walk is a basic stochastic process on graphs and a key primitive in the design of distributed algorithms. One of the most important features of random walks is that, under mild conditions, they converge to a stationary distribution…
Counting the independent sets of a graph is a classical #P-complete problem, even in the bipartite case. We give an exponential-time approximation scheme for this problem which is faster than the best known algorithm for the exact problem.…
In an earlier paper the authors proved that limits of convergent graph sequences can be described by various structures, including certain 2-variable real functions called graphons, random graph models satisfying certain consistency…
In this paper, we give a detailed construction of an example of excited random walk with speed zero in an ergodic random environment that have an infinite average number of cookies in each site. This example confirms that a result of…
The rotor walk on a graph is a deterministic analogue of random walk. Each vertex is equipped with a rotor, which routes the walker to the neighbouring vertices in a fixed cyclic order on successive visits. We consider rotor walk on an…
For a finite graph $G=(V,E)$ let $G^*$ be obtained by considering a random perfect matching of $V$ and adding the corresponding edges to $G$ with weight $\varepsilon$, while assigning weight 1 to the original edges of $G$. We consider…
We introduce the notion of `core growth rate' for strongly synchronizing transducers. We explore some elementary properties of the core growth rate and give examples of transducers with exponential core growth rate. We conjecture that all…