Leaf-to-Leaf Distances and their Moments in Finite and
Infinite m-ary Acyclic Graphs
Andrew M. Goldsborough, S. Alex Rautu and Rudolf A. Römer
Department of Physics and Centre for Scientific Computing, University of Warwick, Coventry, UK, CV4 7AL
Generalization to m-ary trees
Abstract
Periodic trees
It has recently been shown [1] that two-point correlation functions in tensor
network wavefunctions are related to the length of the path through the network
that connects the two lattice sites. In reference to a binary tree tensor network
●
The same logic applies to m-ary trees, except there are m sub-trees of mn-1 leaves
●
Using
●
Hence
●
In the limit
(TTN) [2], we analyse the geometry of a complete regular binary tree where the
leaves represent the points on a 1-D lattice. We find an analytic expression for the
average path distance for a given separation in a tree with n levels. This
expression is then generalised to give any raw statistical moment for m-ary trees
with open and periodic boundary conditions. We also present first results for
, this simplifies to
random binary trees.
Motivation
• Two-point correlation functions in tensor networks are related to the path length
through the network that connects the two sites.
• Only separations of r ≤ 2
n-1
are relevant as larger separations can be described by
smaller separations going the opposite way around the circle. Hence
• Can we derive an analytic expression the average distance for a given separation?
• The binary TTN is the simplest non-trivial tensor network, can this be generalised
• When written in full
to more complex systems? e.g. more higher valencies and incomplete trees [3].
• Can we calculate statistical moments of the distances?
Average path length  (2)n(r) for binary trees
with n = 20 and in the infinite n limit.
Definitions
Level n
Vertex
Root node
Average path length  (m)∞(r) for various
m-ary trees in the infinite limit.
Variance in the path distances
where
Full binary trees
• As the distances (n(r) ) are an average, it is useful to know the spread of the
1
2
values. Variance can be calculated using
3
4
• Defining  (m)n(r) as the sum of the squares of the distances the variance is
Separation r
Leaf
• A binary tree is a connected, 3-valent graph with no loops.
• The root node is the vertex with two degrees that defines the top of the graph. Then
•  (m)n(r) can be found using a recurrence relation
the rest of the vertices have two daughter nodes and one parent.
• A leaf node has no daughters.
• Disordered tree tensor networks [3] have a structure of a full binary tree.
• The depth is the number of vertices from the root node.
• Solving as before gives
• The average path distance requires averaging over all n! possible trees.
• A level (n) is the set of vertices with the same depth.
• We have a computer simulation to generate all trees and count the paths distances.
• The tree is complete if all of the leaf nodes are at the same depth and all the levels
are completely filled.
• We enumerate the leaves from left to right to imitate a lattice. We can then define a
• In full
separation as r = |x1 – x2|.
Binary trees
• Let n(r) be the sum of the paths for a given r,
• Let n(r) be the average distance for a separation r.
Average path distance A(2)1,n(r) for all (L-1)!
full binary trees as a function of separation
r. Error bars are smaller than symbol size.
The grey line is the result of the infinite
complete trees.
• In the infinite n limit
• Tree of n levels can be seen as two n-1 level trees joined by the root
Average path distance A(2)1,n(r) for 500
randomly chosen full binary trees. Error
bars are smaller than line thickness. The
grey line is again the result of the infinite
complete trees.
• Unlike the complete trees the average path distance increases with r until it reaches
a maximum and decreases for large r.
n-1
Future Work
n-1
• Generate a Hamiltonian that has a tensor network in the form of a complete tree to
• When the r > 2n-1 , only need to consider paths that join the two sub-trees, each
analyse how correlation functions relate to the path distance.
such path contributes 2n-1 to n(r).
• When the r ≤ 2n-1 also need the paths within the sub-trees, hence have a recurrence
relation
• Derive an analytic expression for the average path distance in the set of unique,
Variance var[ (2)n(r)] for binary trees with
n = 20 and in the infinite n limit.
Variance var[ (m)∞] for various m-ary trees
in the infinite n limit.
full, binary trees or Catalan Trees.
General moments
• This can be summed over to give
• The kth raw statistical moment is defined as < X k >
• The sum of the paths to the k-th power can easily be written as a recurrence relation
where nc is the critical value for n where the top equation of the recurrence relation
becomes invalid. This is when
• This can then be solved almost as before to
• The sum is just three geometric series' which can be simplified to
Contact Details
• Email: a.goldsborough@warwick.ac.uk, s.a.rautu@warwick.ac.uk,
r.roemer@warwick.ac.uk
• Hence
where
and Φ is the Hurwitz-Lerch Transcendent, defined as the
• Website: http://www.warwick.ac.uk/andrewgoldsborough/
analytic continuation of
References
• In the limit of infinite trees:
• The k-th moment can then be written as
[1] G. Evenbly and G. Vidal, J. Stat. Phys. 145, 891 (2011).
[2] Y. Shi, L. Duan, and G. Vidal, Phys. Rev. A 74, 022320 (2006).
[3] A. M. Goldsborough, and R. A. Römer, arXiv:1401.4874 (2014).
[4] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products. Academic
Press, 1980.