In p.275 of his classical book T. M. Liggett. (1985). Interacting particle systems. Springer], T. M. Liggett remarked that ``The importance of critical exponents is based largely on what is known as the universality principle, which plays an important role in mathematical physics." Here universality principle means that while the value of critical parameter will usually depend on the details of the definition of the model, the value of critical exponent will be the same for large classes of models (called universality classes). This principle has been an important source of problems in mathematical physics and probability theory.

This talk is based on a joint work with Shi Zhan, Sidoravicius Vladas and Wang Longmin, and presents a result on universality of critical exponent for Hausdorff dimensions of boundaries of branching random walks on hyperbolic groups.

Let $\Gamma$ be a nonamenable finitely generated infinite hyperbolic group with a symmetric generating set $S,$ and $\partial\Gamma$ the hyperbolic boundary of its Cayley graph. Fix a symmetric probability $\mu$ on $\Gamma$ whose support is $S,$ and denote by $\rho=\rho(\mu)$ the spectral radius of the random walk $\xi$ on $\Gamma$ associated to $\mu.$ Let $\nu$ be a probability on $\{1,2,3,\cdots\}$ with a finite mean $\lambda.$ Write $\Lambda\subseteq\partial\Gamma$ for the boundary of the branching random walk with offspring distribution $\nu$ and underlying random walk $\xi,$ and $h(\nu)$ for the Hausdorff dimension of $\Lambda.$ When $\lambda>1/\rho,$ the branching random walk is recurrent, trivially

$$\Lambda=\partial\Gamma,\ h(\nu)=\dim(\partial \Gamma).$$

In this talk, we focus on the transient setting i.e. $\lambda\in [1,1/\rho],$ and prove the following results: $h(\nu)$ is a deterministic function of $\lambda$ and thus denote it by $h(\lambda);$ and $h(\lambda)$ is continuous and strictly increasing in $\lambda\in [1,1/\rho]$ and $h(1/\rho)\leq\frac{1}{2}\dim(\partial \Gamma);$ and there is a positive constant $C$ such that

$$h(1/\rho)-h(\lambda)\sim C\sqrt{1/\rho-\lambda}\ \mbox{as}\ \lambda\uparrow 1/\rho.$$

The above results confirm a conjecture of S. Lalley in his ICM 2006 Lecture: The critical exponent for Hausdorff dimensions of boundaries of branching random walks on hyperbolic groups is $1/2$.