Maximal and minimal displacement of supercritical branching random walks on free products of groups

Abstract

We prove that the maximal and minimal displacement of branching random walks with mean offspring number $ρ>1$ on free products of finite groups grows linearly almost surely. More precisely, we establish that the linear speed for the maximal (respectively minimal) displacement is given by the largest (respectively smallest) intersection point of the large deviation rate function of the underlying random walk with the horizontal line at height $\logρ$. The proof is based on constructing an associated multitype branching process which consists of particles that travel fast enough, and distinguishing the types via the suffix of the particles locations.

Figures (4)

• Figure 1: A part of the Cayley graph of the free product $\mathbb{Z}/3\mathbb{Z}*\mathbb{Z}/4\mathbb{Z}$; $a$ is the generator of $\mathbb{Z}/3\mathbb{Z}$ and $b$ is the generator of $\mathbb{Z}/4\mathbb{Z}$, and $S=\{a,a^{-1},b,b^{-1}\}$. Black edges indicates multiplication with $a$ or $a^{-1}$, and blue edges indicate multiplication with $b$ or $b^{-1}$.
• Figure 2: The rate function on $\mathbb{Z}$ with increments ${\mathbb{P}(\xi_1=0)=\mathbb{P}(\xi_1=1)=\frac{1}{2}}$ is $I(x)=x\log(x)+(1-x)\log(1-x)+\log(2)$. The dashed lines indicate that $I$ equals $\infty$ at these values. The drift of the walk is $1/2$, and this is also a root of the rate function.
• Figure 3: The multitype branching process on $\mathbb{Z}/2\mathbb{Z}*\mathbb{Z}/2\mathbb{Z}*\mathbb{Z}/2\mathbb{Z}$: red, blue, and green stand for $1$, $2$ and $3$, the first, second and third copy of $\mathbb{Z}/2\mathbb{Z}$, respectively. We consider all branching random walk particles started in $e$ and stopped after their first exit of $\mathcal{B}_n$. Then we keep only those particles that never left the cone $C(1)$, and reached $\mathcal{B}_n^c$ fast enough. We partition these particles according to their suffix: here we have $Z_{1,1} = 3$, $Z_{1,2} = 3$, and $Z_{1,3} = 4$.
• Figure 4: The two cones $C(1)$ and $C(2)$ in the group $G=\mathbb{Z}/ 2\mathbb{Z}* \mathbb{Z}/ 4\mathbb{Z}$, with $\mathbb{Z}/2\mathbb{Z} = \{e,a\}$ and $\mathbb{Z}/4\mathbb{Z}=\{e,b,b^2,b^3\}$. At each element of $\mathbb{Z}/ 4\mathbb{Z}$ there is a copy of $C(2)$ attached, and at each element of $\mathbb{Z}/2\mathbb{Z}$ there is a copy of $C(1)$ attached.

Theorems & Definitions (34)

• Theorem 1
• Theorem 2
• Example 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4