Up-down chains and scaling limits: application to permuton- and graphon-valued diffusions

TL;DR

This paper identifies some integrable up-down chains and construct their scaling limits, a family of permuton- and graphon-valued Feller diffusions, and exhibits ergodicity, diagonalizable semigroups, and explicit expressions for the maximal separation distance to stationarity.

Abstract

An up-down chain is a Markov chain in which each transition is a two-step process that moves up to a larger object and then back down to an object of the original size. The first goal of this paper is to present a general framework for analyzing these chains and computing their scaling limits. This approach unifies much of the existing literature while extending it in several directions. These include explicit conditions for constructing integrable up-down chains and convergence results for families of intertwined processes. The latter contribute to the method of intertwiners of Borodin and Olshanski. The second goal is to highlight a notable application of this framework to the settings of permutations and graphs. Here, we identify some integrable up-down chains and construct their scaling limits, a family of permuton- and graphon-valued Feller diffusions. Both the up-down chains and the limiting diffusions exhibit ergodicity, diagonalizable semigroups, and explicit expressions for the maximal separation distance to stationarity. For the diffusions, the stationary measures are the recursive separable permutons and recursive cographons recently introduced by the authors, and the separation distances turn out to be related to the Dedekind eta function.

Figures (12)

• Figure 1: Examples of the duplication operations on permutations (left) and graphs (right).
• Figure 2: Simulation of an up-down chain on permutations (left) and graphs (right). In each case, we take $p=1/2$, $n=100$, and the initial distribution is a uniform random permutation, resp. graph, of size $n$. Each movie is a succession of thirty-one pictures that show the state of the chain after $m$ steps, where $m \in \{0,\dots,30\} \cdot 50$. We plot permutations as diagrams (with a blue dot at coordinates $(i,\sigma(i))$ for each $i\le n$), and graphs as pixel pictures, or adjacency matrices (with a black dot at coordinates $(i,j)$ and $(j,i)$ for each edge $\{i,j\}$ of the graph with an appropriate labeling). Animations do not work properly with all pdf viewers, but they seem to work with Acrobat Reader or Okular.
• Figure 3: The separation distance of the permuton- and graphon-valued diffusions.
• Figure 4: Left: the anisotropic Young diagram $\tau=(4,4,3,1,1)$ (for $\theta=.5$) and its interlacing coordinates. Right: the same Young diagram $\tau$, together with two extra boxes, defining diagrams $\lambda$ and $\mu$, each with one more box than $\tau$. We indicated the interlacing coordinates of $\tau$ in which we have inserted an artificial pair $x_1^\tau=y_2^\tau$ for convenience. We also indicated the interlacing coordinates of $\lambda$ and $\mu$ which are different from that of $\tau$. Note that the interlacing coordinates of $\mu$ also contain an artificial pair $x_3^\mu=y_4^\mu$. These coordinates satisfy \ref{['eq:xj1']} and \ref{['eq:xj2']}.
• Figure 5: The $\theta$-embedding of a Young diagram $\lambda$ in $\Omega$. The $a_i$'s and $b_i$'s are the areas of the colored regions of the diagram.

Theorems & Definitions (121)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 1.4
• Remark 1.5
• Theorem 2.1: Chapter 4 Theorem 2.7 in ethierKurtzBook
• Proposition 2.2: Proposition 19.9 in kallenbergTheBible
• Theorem 2.3: Chapter 4, Theorem 2.12 in ethierKurtzBook
• Theorem 2.4: Chapter 1, Theorems 6.1, 6.5 in ethierKurtzBook
• Proposition 2.5