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Maximal entropy random walks and central Markov chains

Yoann Offret, Sergey Dovgal

TL;DR

This work develops Maximal Entropy Random Walks on Weighted Bratteli Diagrams, recasting MERWs as limit central Markov chains and connecting them to non-negative harmonic functions and Martin boundary theory. It unifies finite irreducible graphs with countable Bratteli diagrams, and applies the framework to growth models such as prefix trees, the Young lattice, and various $d$-ary/tree-like structures, yielding new proofs and generalizations of hook-length identities and Plancherel-type limit shapes. The paper further links MERWs to central measures for aggregation processes, the Chinese restaurant process, and Poisson–Dirichlet limits, while proposing a Knuth-based Monte Carlo algorithm to approximate MERWs in complex BD settings. Together, these results provide a versatile toolkit for deriving combinatorial identities, asymptotics of growth processes, and practical estimation methods for entropy-maximizing random walks on hierarchical networks.

Abstract

We introduce and develop the concept of Maximal Entropy Random Walks (MERWs) on Weighted Bratteli Diagrams (WBDs), maximizing entropy production along paths as a natural criterion for choosing random walks on networks. Initially defined for irreducible finite graphs, MERWs were recently extended to the infinite setting in [1]. Bratteli Diagrams model various growth processes, such as the Young Lattice, where the Plancherel growth process emerges as a MERW. We show that MERWs are special cases of central Markov chains, which, in general, provide a powerful framework for deriving combinatorial identities. Regarding growing trees, in particular, we retrieve and extend Han's hook-length formula for binary trees and demonstrate that the Binary Search Tree (BST) process is a MERW, recovering its asymptotic behavior. We also introduce preferential attachment to generalize BSTs. For comb models, significant central measures appear, including the Chinese restaurant process, providing an alternative proof of the Poisson-Dirichlet limit distribution. Finally, we propose a Monte Carlo method, based on Knuth's algorithm, to approximate MERWs. We apply it to a pyramidal growth model, drawing connections with the limit shape of Young diagrams under the Plancherel measure.

Maximal entropy random walks and central Markov chains

TL;DR

This work develops Maximal Entropy Random Walks on Weighted Bratteli Diagrams, recasting MERWs as limit central Markov chains and connecting them to non-negative harmonic functions and Martin boundary theory. It unifies finite irreducible graphs with countable Bratteli diagrams, and applies the framework to growth models such as prefix trees, the Young lattice, and various -ary/tree-like structures, yielding new proofs and generalizations of hook-length identities and Plancherel-type limit shapes. The paper further links MERWs to central measures for aggregation processes, the Chinese restaurant process, and Poisson–Dirichlet limits, while proposing a Knuth-based Monte Carlo algorithm to approximate MERWs in complex BD settings. Together, these results provide a versatile toolkit for deriving combinatorial identities, asymptotics of growth processes, and practical estimation methods for entropy-maximizing random walks on hierarchical networks.

Abstract

We introduce and develop the concept of Maximal Entropy Random Walks (MERWs) on Weighted Bratteli Diagrams (WBDs), maximizing entropy production along paths as a natural criterion for choosing random walks on networks. Initially defined for irreducible finite graphs, MERWs were recently extended to the infinite setting in [1]. Bratteli Diagrams model various growth processes, such as the Young Lattice, where the Plancherel growth process emerges as a MERW. We show that MERWs are special cases of central Markov chains, which, in general, provide a powerful framework for deriving combinatorial identities. Regarding growing trees, in particular, we retrieve and extend Han's hook-length formula for binary trees and demonstrate that the Binary Search Tree (BST) process is a MERW, recovering its asymptotic behavior. We also introduce preferential attachment to generalize BSTs. For comb models, significant central measures appear, including the Chinese restaurant process, providing an alternative proof of the Poisson-Dirichlet limit distribution. Finally, we propose a Monte Carlo method, based on Knuth's algorithm, to approximate MERWs. We apply it to a pyramidal growth model, drawing connections with the limit shape of Young diagrams under the Plancherel measure.

Paper Structure

This paper contains 31 sections, 26 theorems, 164 equations, 17 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Irreducible framework
  3. Beyond the irreducible framework : Bratteli Diagrams.
  4. MERWs on WBDs
  5. WBDs
  6. Random Walks and Entropy maximization
  7. A consistent definition
  8. MERWs as particular central Markov chains.
  9. Connection with MERWs on irreducible graphs
  10. Illustrative examples
  11. Maximal entropy growing trees
  12. The BD structure of $\mathcal{P}$
  13. Underlying skeleton associated with a boundary point
  14. Correspondence with fragmentation processes
  15. About MERWs
  16. ...and 16 more sections

Key Result

Lemma 2.1

For any non-negative integer $n$, the distribution $\mu_n$ belongs to $\mathcal{RW}_n$. More precisely, the corresponding transition kernel is given for all $x \nearrow y$ by Furthermore, for all $s \in \mathcal{T}_n$, one can write Additionally, one has $\varphi_n(\varnothing) = 1$ and for all $x$ with $|x| < n$,

Figures (17)

  • Figure 1: An example of Brattelli Diagram with some transition probabilities
  • Figure 2: The first example, on the left, show that the unique MERW has no full support. On the the second one, it is represented the transition probabilities of the three possible MERWs.
  • Figure 3: Weighted Pascal lattice
  • Figure 4: A standard Young tableau (in French notation) representing a path in $\mathbb{Y}$ from the root to the partition $8+5+4+1=18$.
  • Figure 5: Modified Frobenius coordinates of a Young diagram (in Russian style). For example, $g_1 = 1/2$ and $f_1 = 3/2$.
  • ...and 12 more figures

Theorems & Definitions (76)

  • Remark 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.1
  • Lemma 2.1
  • Remark 2.2
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 66 more