Mixing of fast random walks on dynamic random permutations

TL;DR

The paper investigates a fast random walk on dynamic random permutations generated by random transpositions, comparing coagulative dynamics (no fragmentation) and coagulative-fragmentative dynamics (allow fragmentation). On the scale $t\sim n$, the total variation distance to equilibrium converges to a limit process that drops once at a random drop-down time and then follows a deterministic curve linked to Erdős-Rényi giant-component sizes; the jump time distribution differs between the two dynamics. For CDP, the drop-down time converges with $\mathbb{P}(s^{\Downarrow}\le s)=\eta(s)$ and the mixing profile is $\mathcal{D}_n^{v_0}(sn) \to 1-\eta(s)\mathbb{1}_{\{}\}$, while for CFDP the drop-down time converges with $\mathbb{P}(u^{\Downarrow}\le u)=\zeta(u)$ and the mixing profile is $\mathcal{D}_n^{v_0}(un) \to 1-\zeta(u)\mathbb{1}_{\{\}}$. After the drop-down, local mixing occurs rapidly on the giant component, ensuring a single macroscopic drop in the total-variation distance and a universal decay toward equilibrium. The analysis leverages couplings between coagulative/fragementative dynamics and ER graph processes, notably cycle-free ER and Schramm’s coupling, to link permutation-cycle structure with ER giant-component behavior and to obtain precise limit laws and process-level convergence in the Skorokhod $M_1$ topology.

Abstract

We analyse the mixing profile of a random walk on a dynamic random permutation, focusing on the regime where the walk evolves much faster than the permutation. Two types of dynamics generated by random transpositions are considered: one allows for coagulation of permutation cycles only, the other allows for both coagulation and fragmentation. We show that for both types, after scaling time by the length of the permutation and letting this length tend to infinity, the total variation distance between the current distribution and the uniform distribution converges to a limit process that drops down in a single jump. This jump is similar to a one-sided cut-off, occurs after a random time whose law we identify, and goes from the value 1 to a value that is a strictly decreasing and deterministic function of the time of the jump, related to the size of the largest component in Erdős-Rényi random graphs. After the jump, the total variation distance follows this function down to 0.

Figures (9)

• Figure 1: The red curve is a typical evolution of the total variation distance for an infinitely fast random walk on a coagulative dynamic permutation. The blue curve is a plot of the deterministic function of the scaled time to which the total variation distance drops at a random time and subsequently sticks to.
• Figure 2: Simulations of the evolution of the total variation distance for $10^2$ different realisations of a coagulative dynamic permutation of $10^4$ elements and an infinitely fast random walk on top. Each simulation run corresponds to a single coloured curve.
• Figure 3: The same as Fig. \ref{['fig:intro']} for a coagulative-fragmentative dynamic permutation.
• Figure 4: The same as Fig. \ref{['fig:sim-coag']} for a coagulative-fragmentative permutation.
• Figure 5: Example of an evolution of an ISRW on top of a CFDP with three elements starting from the identity permutation. The first row shows the transpositions that generate the next permutation. The second row is a graphical representation of the cycles of this permutation. The third row shows the evolution of the ISRW distribution, given that it started from the element $1$.

Theorems & Definitions (76)

• Definition 1.1: Dynamic permutation
• Example 1.2: Transpositions may fragment cycles or coagulate cycles
• Definition 1.3: Coagulative dynamic permutation
• Definition 1.4: Coagulative-fragmentative dynamic permutation
• Remark 1.5: Time horizon for dynamic permutations and cycle structure
• Definition 1.6: Infinite-speed random walk on $\Pi_n$
• Definition 1.7: Standard Erdős-Rényi multi-graph process
• Remark 1.8: Versions and asymptotic equivalence
• Definition 1.9: Cycle-free Erdős-Rényi graph process
• Definition 1.10: Functions related to the structure of Erdős-Rényi random graphs