Infinite weighted graphs with bounded resistance metric

  • Palle Jorgensen
  • Feng Tian

Abstract

We consider infinite weighted graphs $G$, i.e., sets of vertices $V$, and edges $E$ assumed countably infinite. An assignment of weights is a positive symmetric function $c$ on $E$ (the edge-set), conductance. From this, one naturally defines a reversible Markov process, and a corresponding Laplace operator acting on functions on $V$, voltage distributions. The harmonic functions are of special importance. We establish explicit boundary representations for the harmonic functions on $G$ of finite energy.

We compute a resistance metric $d$ from a given conductance function. (The resistance distance $d(x,y)$ between two vertices $x$ and $y$ is the voltage drop from $x$ to $y$, which is induced by the given assignment of resistors when $1$ amp is inserted at the vertex $x$, and then extracted again at $y$.)

We study the class of models where this resistance metric is bounded. We show that then the finite-energy functions form an algebra of ${1}/{2}$-Lipschitz-continuous and bounded functions on $V$, relative to the metric $d$. We further show that, in this case, the metric completion $M$ of $(V,d)$ is automatically compact, and that the vertex-set $V$ is open in $M$. We obtain a Poisson boundary-representation for the harmonic functions of finite energy, and an interpolation formula for every function on $V$ of finite energy. We further compare $M$ to other compactifications; e.g., to certain path-space models.

References

Albeverio, S. and Kusuoka, S., Diffusion processes in thin tubes and their limits on graphs, Ann. Probab. 40 (2012), no. 5, 2131–2167. https://doi.org/10.1214/11-AOP667

>

Alpay, D. and Jorgensen, P., Reproducing kernel Hilbert spaces generated by the binomial coefficients, Illinois J. Math. 58 (2014), no. 2, 471–495.

Alpay, D., Jorgensen, P., Lewkowicz, I., and Marziano, I., Representation formulas for Hardy space functions through the Cuntz relations and new interpolation problems, in “Multiscale signal analysis and modeling”, Springer, New York, 2013, pp. 161--182. https://doi.org/10.1007/978-1-4614-4145-8_7

>

Alpay, D., Jorgensen, P., Seager, R., and Volok, D., On discrete analytic functions: products, rational functions and reproducing kernels, J. Appl. Math. Comput. 41 (2013), no. 1-2, 393–426. https://doi.org/10.1007/s12190-012-0608-2

>

Alpay, D., Jorgensen, P., and Volok, D., Relative reproducing kernel Hilbert spaces, Proc. Amer. Math. Soc. 142 (2014), no. 11, 3889–3895. https://doi.org/10.1090/S0002-9939-2014-12121-6

>

Ancona, A., Théorie du potentiel sur les graphes et les variétés, in “École d'été de Probabilités de Saint-Flour XVIII—1988”, Lecture Notes in Math., vol. 1427, Springer, Berlin, 1990, pp. 1--112. https://doi.org/10.1007/BFb0103041

>

Bayer, C. and Veliyev, B., Utility maximization in a binomial model with transaction costs: a duality approach based on the shadow price process, Int. J. Theor. Appl. Finance 17 (2014), no. 4, 1450022, 27 pp. https://doi.org/10.1142/S0219024914500228

>

Bezuglyi, S., Kwiatkowski, J., and Yassawi, R., Perfect orderings on finite rank Bratteli diagrams, Canad. J. Math. 66 (2014), no. 1, 57–101. https://doi.org/10.4153/CJM-2013-041-6

>

Bratteli, O., Inductive limits of finite dimensional $C^ast $-algebras, Trans. Amer. Math. Soc. 171 (1972), 195–234. https://doi.org/10.2307/1996380

>

Bratteli, O., Jorgensen, P. E. T., Kim, K. H., and Roush, F., Non-stationarity of isomorphism between AF algebras defined by stationary Bratteli diagrams, Ergodic Theory Dynam. Systems 20 (2000), no. 6, 1639–1656. https://doi.org/10.1017/S0143385700000912

>

Bratteli, O., Jorgensen, P. E. T., and Ostrovs'kyu ı, V., Representation theory and numerical AF-invariants. The representations and centralizers of certain states on $mathscr O_d$, Mem. Amer. Math. Soc. 168 (2004), no. 797, xviii+178. https://doi.org/10.1090/memo/0797

>

Chang, X., Xu, H., and Yau, S.-T., Spanning trees and random walks on weighted graphs, Pacific J. Math. 273 (2015), no. 1, 241–255. https://doi.org/10.2140/pjm.2015.273.241

>

Doob, J. L., The structure of a Markov chain, in “Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory”, Univ. California Press, Berkeley, Calif., 1972, pp. 131--141.

Dunford, N. and Schwartz, J. T., Linear operators. Part II. spectral theory. selfadjoint operators in hilbert space, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988, reprint of the 1963 original.

Dutkay, D. E. and Jorgensen, P. E. T., Martingales, endomorphisms, and covariant systems of operators in Hilbert space, J. Operator Theory 58 (2007), no. 2, 269–310.

Dutkay, D. E. and Jorgensen, P. E. T., Affine fractals as boundaries and their harmonic analysis, Proc. Amer. Math. Soc. 139 (2011), no. 9, 3291–3305. https://doi.org/10.1090/S0002-9939-2011-10752-4

>

Dutkay, D. E., Jorgensen, P. E. T., and Silvestrov, S., Decomposition of wavelet representations and Martin boundaries, J. Funct. Anal. 262 (2012), no. 3, 1043–1061. https://doi.org/10.1016/j.jfa.2011.10.010

>

Georgakopoulos, A., Haeseler, S., Keller, M., Lenz, D., and Wojciechowski, R. K., Graphs of finite measure, J. Math. Pures Appl. (9) 103 (2015), no. 5, 1093–1131. https://doi.org/10.1016/j.matpur.2014.10.006

>

Giordano, T., Putnam, I. F., and Skau, C. F., Full groups of Cantor minimal systems, Israel J. Math. 111 (1999), 285–320. https://doi.org/10.1007/BF02810689

>

Gorodezky, I. and Pak, I., Generalized loop-erased random walks and approximate reachability, Random Structures Algorithms 44 (2014), no. 2, 201–223. https://doi.org/10.1002/rsa.20460

>

Herman, R. H., Putnam, I. F., and Skau, C. F., Ordered Bratteli diagrams, dimension groups and topological dynamics, Internat. J. Math. 3 (1992), no. 6, 827–864. https://doi.org/10.1142/S0129167X92000382

>

Hersonsky, S., Boundary value problems on planar graphs and flat surfaces with integer cone singularities, I: The Dirichlet problem, J. Reine Angew. Math. 670 (2012), 65–92.

Jorgensen, P. E. T., A uniqueness theorem for the Heisenberg-Weyl commutation relations with nonselfadjoint position operator, Amer. J. Math. 103 (1981), no. 2, 273–287. https://doi.org/10.2307/2374217

>

Jorgensen, P. E. T., Essential self-adjointness of the graph-Laplacian, J. Math. Phys. 49 (2008), no. 7, 073510, 33 pp. https://doi.org/10.1063/1.2953684

>

Jorgensen, P. E. T., A sampling theory for infinite weighted graphs, Opuscula Math. 31 (2011), no. 2, 209–236. https://doi.org/10.7494/OpMath.2011.31.2.209

>

Jorgensen, P. E. T. and Pearse, E. P. J., A Hilbert space approach to effective resistance metric, Complex Anal. Oper. Theory 4 (2010), no. 4, 975–1013. https://doi.org/10.1007/s11785-009-0041-1

>

Jorgensen, P. E. T. and Pearse, E. P. J., Resistance boundaries of infinite networks, in “Random walks, boundaries and spectra”, Progr. Probab., vol. 64, Birkhäuser/Springer Basel AG, Basel, 2011, pp. 111--142. https://doi.org/10.1007/978-3-0346-0244-0_7

>

Keller, M. and Lenz, D., Dirichlet forms and stochastic completeness of graphs and subgraphs, J. Reine Angew. Math. 666 (2012), 189–223. https://doi.org/10.1515/CRELLE.2011.122

>

Kostrykin, V., Potthoff, J., and Schrader, R., Brownian motions on metric graphs, J. Math. Phys. 53 (2012), no. 9, 095206, 36 pp. https://doi.org/10.1063/1.4714661

>

Roblin, T., Comportement harmonique des densités conformes et frontière de Martin, Bull. Soc. Math. France 139 (2011), no. 1, 97–128.

Rudin, W., Real and complex analysis, third ed., McGraw-Hill Book Co., New York, 1987.

Rudin, W., Functional analysis, second ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.

Sawyer, S. A., Martin boundaries and random walks, in “Harmonic functions on trees and buildings (New York, 1995)'', Contemp. Math., vol. 206, Amer. Math. Soc., Providence, RI, 1997, pp. 17--44. https://doi.org/10.1090/conm/206/02685

>

Skopenkov, M., The boundary value problem for discrete analytic functions, Adv. Math. 240 (2013), 61–87. https://doi.org/10.1016/j.aim.2013.03.002

>

Tosiek, J. and Brzykcy, P., States in the Hilbert space formulation and in the phase space formulation of quantum mechanics, Ann. Physics 332 (2013), 1–15. https://doi.org/10.1016/j.aop.2013.01.010

>

Wojciechowski, R. K., Stochastic completeness of graphs, Ph.D. Theses, eprint arxiv:0712.1570 [math.SP], 2007.

Published
2018-08-01
How to Cite
Jorgensen, P., & Tian, F. (2018). Infinite weighted graphs with bounded resistance metric. MATHEMATICA SCANDINAVICA, 123(1), 5-38. https://doi.org/10.7146/math.scand.a-106208
Section
Articles