Journal of Pure and Applied Mathematics: Advances and ApplicationsAuthor's: A. M. Goldsborough, J. M. Fellows, S. A. Rautu, M. Bates, G. Rowlands and R. A. Römer
Pages: [1] - [29]
Received Date: April 13, 2018; Revised June 2, 2018
Submitted by:
DOI: http://dx.doi.org/10.18642/jpamaa_7100121956
Tree tensor networks are extensively used for both computational and analytic study of strongly correlated quantum matter, in which their path lengths are related to correlation functions. In this context, we study the properties of leaf-to-leaf distances on ordered Catalan tree graphs by using a diagrammatic method. This allows us to obtain an explicit analytic formula for the generating function of the leaf-to-leaf path lengths. By using this function, the statistical moments of such path lengths can be computed exactly. Here, we show how to construct the average distances as a function of separation of the leaves from the generating function, and study their asymptotic properties. Moreover, we find that these are equivalent to the average length of paths starting from the root node, as expected by the rerooting invariance of a tree at a random vertex.
Catalan trees, path lengths, diagrammatic representation.
