Infinite cycles in the interchange process in five dimensions

Abstract

In the interchange process on a graph $G=(V,E)$, distinguished particles are placed on the vertices of $G$ with independent Poisson clocks on the edges. When the clock of an edge rings, the two particles on the two sides of the edge interchange. In this way, a random permutation $π_β:V\to V$ is formed for any time $β>0$. One of the main objects of study is the cycle structure of the random permutation and the emergence of long cycles. We prove the existence of infinite cycles in the interchange process on $\mathbb Z ^d$ for all dimensions $d\ge 5$ and all large $β$, establishing a conjecture of Bálint Tóth from 1993 in these dimensions. In our proof, we study a self-interacting random walk called the cyclic time random walk. Using a multiscale induction we prove that it is diffusive and can be coupled with Brownian motion. One of the key ideas in the proof is establishing a local escape property which shows that the walk will quickly escape when it is entangled in its history in complicated ways.

Figures (4)

• Figure 1: A cyclic time random walk on $\mathbb Z$. The red bars indicate a ring on the corresponding edge. The pattern repeats periodically when time is shifted up by $\beta$. The blue path is the cyclic time random walk $W(t)$ starting from the origin. When the walk meets a horizontal bar it jumps across it to the other side. In this example the walk closes at time $3\beta$ and therefore the cycle of the origin is of length $3$. Note also that $-1$ is a fixed point of $\pi _\beta$ and $(2,3)$ is a cycle of length $2$.
• Figure 2: The escape algorithm. Corollary \ref{['cor:A']} shows that with very high probability at least one of the red paths will escape far. Most likely, this is true after conditioning on the history before entering $B$ (the purple path). Once we enter $B$, we reveal the red paths and find the successful starting point ($p_2$). Then, we use Lemma \ref{['lem:connect']} to connect $v$ to this starting point at the right time and escape. For the proof of Lemma \ref{['lem:connect']}, we first use Claim \ref{['claim:green']} that shows that with positive probability, the green path will avoid the red paths and connect to $B^-$ in a short amount of time. Then, we use Claim \ref{['claim:chain']} to connect $v$ to $w$ while staying in $B^-$. This is done by splitting the blue path into $O(\log n)$ many paths of different length scales and use the Brownian approximation in each scale.
• Figure 3: Heavy blocks. We wish to show that a block $B$ of side length $2^m\ge t_n^{\delta }$ is not heavy. Every time we enter a small blue block $B'\in \mathcal{B}_j$ inside $B$ for the first time, by the escape algorithm, we have a chance of at least $n^{-\gamma }$ to escape fast from $B$ and not return for a long time. If the escape attempt failed and we returned to $B$ after time $n4^m$ then we try again with a new small blue block. With very high probability, this should not happen more than $n^{\gamma +3}$ times which is not enough in order to make $B$ heavy.
• Figure 4: The proof of Lemma \ref{['l:relaxed.pts']}. We wish to show that along the blue path, most of the times $s\in [t-t_k',t]$ are relaxed. The blocks $B(W(s),r)$ for $r\ge t_n^\delta$ are not heavy by Lemma \ref{['lem:heavy2']} and therefore it suffices to show that the blocks $B(W(s),r)$ for $r\le t_n^\delta$ are not heavy for most times $s$. First, we claim that for most of the times $s$, the earlier history of the walk before $t-t_k'$ (the red paths in the picture) is at distance at least $t_n^\delta$ away from $W(s)$. Indeed, by Claim \ref{['claim:cover']}, there are at most $n^C$ paths such that the earlier history is contained in a small neighbourhood of these paths. By the pair property, most of the time the blue path will be far from all of them and therefore far from the entire history before $t-t_k'$. Thus, we only have to show that for most times $s$ the blocks $B(W(s),r)$ for $r\le t_n^\delta$ are not heavy with respect to the blue path. This follows from the fact that $B$ is good and therefore the blue path is relaxed.

Theorems & Definitions (76)

• Conjecture : Tóth 1993
• Theorem 1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Claim 2.3
• proof
• Definition 3.1
• Theorem 3.2