Infinite excursions of rotor walks on regular trees

Sebastian Mueller; Tal Orenshtein

Home/Archive/arXiv

arXiv · 1511.05896

Infinite excursions of rotor walks on regular trees

Sebastian Mueller·Tal Orenshtein

View original source ↗Read source document ↗Export metadata ↓

Abstract

A rotor configuration on a graph contains in every vertex an infinite ordered sequence of rotors, each is pointing to a neighbor of the vertex. After sampling a configuration according to some probability measure, a rotor walk is a deterministic process: at each step it chooses the next unused rotor in its current location, and uses it to jump to the neighboring vertex to which it points. Rotor walks capture many aspects of the expected behavior of simple random walks. However, this similarity breaks down for the property of having an infinite excursion. In this paper we study that question for natural random configuration models on regular trees. Our results suggest that in this context the rotor model behaves like the simple random walk unless it is not "close to" the standard rotor-router model.

Explore related subjects

math.PR math.CO

Keep this discovery

BibTeX RIS

Sebastian Mueller, Tal Orenshtein. 2017-07-04. Infinite excursions of rotor walks on regular trees. https://arxiv.org/abs/1511.05896

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Averaging principles for nonautonomous multiscale stochastic Burgers equations with reflection

In this paper, we study averaging principles for nonautonomous multiscale stochastic Burgers equations with reflection. First, we derive a general averaging principle applicable to such equations under minimal assumptions. Subsequently, since the coefficients of the obtained averaged equation still depend on the small scaling parameter $\e$, we impose either periodic or asymptotic conditions on the coefficients, thereby obtain two distinct averaged equations whose coefficients are independent of $\e$ and establish two averaging principles. Stopping times and Khasminskii's time discretization schemes play an important role. Finally, a concrete example is provided to illustrate the applicability and validity of the theoretical results.

math.PR↗

arXiv2026

Spectral properties of Random Matrices

We give the theoretical foundations of random matrix theory through the definitions of a random matrix, a random probability measure and the corresponding empirical spectral distribution. The technical tool we use is the Stieltjes transform method through which we prove optimal convergence of the empirical spectral distribution of random sample covariance matrices to the deterministic Marchenko-Pastur distribution. We also give new results about the rigidity of the eigenvalues of this random sample covariance matrix and the rate of their convergence. We then define the Dyson equation method to prove new local laws about a random matrix model that interpolates between the Marchenko-Pastur distribution, the elliptical law and the circular law. Through our work these local laws can be considered universal.

math.PR↗

arXiv2026

Moments approach for the elephant random walk

We discuss the method of moments for the one-dimensional elephant random walk (ERW). We first derive a differential recurrence relation for the characteristic function of the ERW, which yields a corresponding system of recurrence relations for its moments. We then obtain asymptotic approximations for the moments in each of the three parameter regimes of the ERW. Finally, by establishing the convergence of the moments and verifying the corresponding moment-determinacy conditions, we identify the limiting distributions of the ERW in each regime.

math.PR↗