[Paper Review] Existence of a phase transition of the interchange process on the Hamming graph

Batı Şengül, Piotr Miłoś|arXiv (Cornell University)|May 11, 2016

Stochastic processes and statistical mechanics13 references3 citations

TL;DR

This paper establishes a phase transition in the interchange process on the 2D Hamming graph $H(2,n)$, showing that for $t = \beta n^2$, when $\beta > 1/2$, a positive density of vertices lie in cycles of length at least $n^{2-\varepsilon}$ for any $\varepsilon > 0$, while for $\beta < 1/2$, all cycles are $O(\log n)$. The authors develop novel techniques to control cycle concentration on rows and columns, enabling a rigorous analysis of cycle growth via a random graph process and coupling with percolation.

ABSTRACT

The interchange process on a finite graph is obtained by placing a particle on each vertex of the graph, then at rate 1, selecting an edge uniformly at random and swapping the two particles at either end of this edge. In this paper we develop new techniques to show the existence of a phase transition of the interchange process on the 2-dimensional Hamming graph. We show that in the subcritical phase, all of the cycles of the process have length $O(\log n)$, whereas in the supercritical phase a positive density of vertices lie in cycles of length at least $n^{2-\varepsilon}$ for any $\varepsilon&gt;0$.

Motivation & Objective

• To establish the existence of a phase transition in the interchange process on the 2D Hamming graph $H(2,n)$, where cycle lengths shift from logarithmic to macroscopic scale.
• To overcome the limitations of applying techniques from the complete graph to the Hamming graph, which has a different geometric structure.
• To develop new methods to prevent cycles from concentrating on single rows or columns, a key obstacle in extending results from complete graphs.
• To show that in the supercritical phase ($\beta > 1/2$), a positive density of vertices belong to cycles of size $n^{2-\varepsilon}$ for any $\varepsilon > 0$, approaching macroscopic size.
• To lay the groundwork for extending results to more general graphs, including $\mathbb{Z}^d$ and higher-dimensional Hamming graphs.

Proposed method

• Introduce a random graph process $G^t_s$ where edges are added over time based on particle swaps in the interchange process, tracking the growth of cycle components.
• Use a coupling argument with Bernoulli percolation to establish the subcritical phase, showing that with high probability, no cycles exceed $O(\log n)$ length.
• Apply a sprinkling argument to show that the giant component in $G^t_s$ emerges rapidly—within $n^{2-\alpha_1}\log n$ time—implying rapid growth of large cycles.
• Establish bounds on the expected number of vertices in small cycles using moment estimates and Markov’s inequality, controlling the number of vertices that split into small cycles.
• Use induction over time intervals $[t - \Delta_h, t]$ to iteratively prove the existence of cycles of increasing size $n^{\alpha_h}$, with positive density, under the supercritical condition.
• Develop a novel control mechanism to prevent cycles from concentrating on any single row or column, using estimates on the intersection of cycles with lines and exponential tail bounds on cycle lengths.

Experimental results

Research questions

• RQ1 Does the interchange process on the 2D Hamming graph exhibit a phase transition, with a sharp change in cycle length distribution at a critical time $t = \beta n^2$?
• RQ2 Can cycle concentration on rows or columns be controlled to prevent the formation of long cycles that are confined to one-dimensional structures?
• RQ3 What is the maximal cycle size achievable in the supercritical phase, and how close can it get to macroscopic size ($\sim n^2$)?
• RQ4 Can the techniques used for the complete graph be adapted to graphs with different geometry, such as the Hamming graph?
• RQ5 Is it possible to refine the current bounds to show the existence of cycles of size $\Omega(n^2)$ in the supercritical phase?

Key findings

• For $\beta < 1/2$, the subcritical phase is established: $\lim_{n\to\infty}\mathbb{P}(|V_t(C\log n)| = 0) = 1$, meaning all cycles are $O(\log n)$ in length with high probability.
• For $\beta > 1/2$, the supercritical phase is proven: $\lim_{n\to\infty}\mathbb{P}(|V_t(n^{2-\varepsilon})| \geq Cn^2) = 1$ for any $\varepsilon > 0$, indicating a positive density of vertices lie in cycles of size at least $n^{2-\varepsilon}$.
• The authors show that in the supercritical phase, for any $\Delta_n \geq n^{1/2}\log^3 n$, $\lim_{n\to\infty}\mathbb{P}(\sup_{u\in[t-\Delta_n,t]}|V_u(n^2/\log^4 n)| \geq Cn^2) = 1$, indicating a giant component of large cycles persists over a time interval.
• The paper proves that cycles cannot concentrate on any single row or column with high probability, a key technical innovation enabling the phase transition result.
• The authors conjecture that the $\log^4 n$ factor in the cycle size bound can be improved to polylogarithmic or even constant, suggesting the existence of true macroscopic cycles in the supercritical phase.
• The results are expected to extend to higher-dimensional Hamming graphs $H(d,n)$ for fixed $d$, and the authors conjecture a similar phase transition for $H(n,2)$ (the hypercube) with a critical time $t = \beta dk^d / (2d-2)$.