Monopoles, Dipoles, and Harmonic Functions on Bratteli Diagrams

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Monopoles, Dipoles, and Harmonic Functions on Bratteli Diagrams Sergey Bezuglyi1,2

· Palle E.T. Jorgensen3

Received: 2 August 2016 / Accepted: 16 May 2018 / Published online: 31 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018

Abstract In our study of electrical networks we develop two themes: finding explicit formulas for special classes of functions defined on the vertices of a transient network, namely monopoles, dipoles, and harmonic functions. Secondly, our interest is focused on the properties of electrical networks supported on Bratteli diagrams. We show that the structure of Bratteli diagrams allows one to describe algorithmically harmonic functions as well as monopoles and dipoles. We also discuss some special classes of Bratteli diagrams (stationary, Pascal, trees), and we give conditions under which the harmonic functions defined on these diagrams have finite energy. Keywords Bratteli diagram · Laplace operator · Random walk · Electrical network · Monopole · Dipole · Harmonic function · Semibranching function system · Pascal graph · Green’s function · Symmetry Mathematics Subject Classification 37B10 · 37L30 · 47L50 · 60J45

1 Introduction The interest in discrete harmonic analysis includes both classical roots, as well as new and recent research directions. But for both the classical and more recent vintage papers, the question of explicit formulas for the harmonic functions, and their finite energy properties, seems to not yet have received systematic attention. While there are many approaches, our To the memory of Ola Bratteli.

B S. Bezuglyi

[email protected]; [email protected] P.E.T. Jorgensen [email protected]

1

Department of Mathematics, Institute for Low Temperature Physics, Kharkiv 61103, Ukraine

2

Present address: Department of Mathematics, University of Iowa, Iowa City, 52242, IA, USA

3

Department of Mathematics, University of Iowa, Iowa City, 52242, IA, USA

170

S. Bezuglyi, P.E.T. Jorgensen

present paper focuses on one in particular. To make the results more relevant, we further narrow our focus to a particular class of discrete structures, infinite sets of vertices V and edges E. We will consider infinite graphs that can be represented as Bratteli diagrams, a kind of graded graphs with some specific properties (see Definition 3.1). It turns out that for Bratteli diagrams harmonic functions can be algorithmically determined. Our paper is motivated by a number of loosely connected areas of mathematics, all related to a framework of discrete harmonic analysis. On the one hand, it has a C ∗ -algebraic component via our use of Bratteli diagrams, see e.g., [10–13]. In this case, the harmonic analysis questions have amenable answers, but, outside the case of Bratteli diagrams with constant incidence matrix, the most natural classification question is known to be undecidable; see the cited papers. Since their inception in 1972, Bratteli diagrams have found a host of powerful applications, both to many diverse areas within mathematics, as well