Modern theory of electrical networks: from matrix tree theorem to the theory of cluster varieties

B. S. Bychkov^ab, A. A. Kazakov^bcd, D. V. Talalaev<sup>ec

a</sup> University of Haifa, Haifa, Israel
^b National Research University Higher School of Economics
^c P.G. Demidov Yaroslavl State University
^d Lobachevsky Institute of Mathematics and Mechanics, Kazan Federal University
^e Lomonosov Moscow State University

Abstract: The theory of electrical networks, in its current state, covers several areas of contemporary mathematics and mathematical physics, such as the combinatorics of paths, forests, and woods on graphs, discrete harmonic analysis, problems of random walks, exactly solved models in statistical physics, cluster varieties related to spaces of totally positive matrices, discrete integrable systems, algebraic structures similar to Zamolodchikov's tetrahedron equation, and many others. The main aim of the survey is presenting some of these subjects, classical and recently discovered alike.

Keywords: electrical networks, graphs, tree theorem, Ising model, Postnikov model, Lam embedding, Grassmannians, cluster varieties, electrical impedance tomography, phylogenetic networks, random walks.

UDC: 515.1+512+519.1

Received: 01.08.2025

DOI: 10.4213/rm10269