报告人：向开南 教授（湘潭大学）

时间：4月22日（周一）上午10：00

地点：管理科研楼1308

摘要：

In p.275 of his classical book [T. M. Liggett. (1985).Interacting particle systems. Springer], T. M. Liggett remarked that “Theimportance of critical exponents is based largely on what is known as theuniversality principle, which plays an important role in mathematical physics.”Here universality principle means that while the value of critical parameterwill usually depend on the details of the definition of the model, the value ofcritical exponent will be the same for large classes of models (called universalityclasses). This principle has been an important source of problems inmathematical physics and probability theory.
This talk is based on a joint work with Shi Zhan, Sidoravicius Vladas and WangLongmin, and presents a result on universality of critical exponent forHausdorff dimensions of boundaries of branching random walks on hyperbolic groups.
Let Γ be a nonamenable finitely generated infinite hyperbolic group with asymmetric generating set S, and ∂Γ the hyperbolic boundary of its Cayley graph.Fix a symmetric probability µ on Γ whose support is S, and denote by ρ = ρ(µ)the spectral radius of the random walk ξ on Γ associated to µ. Let ν be aprobability on {1,2,3, ・ ・ ・ } with a finite mean λ. Write Λ ⊆ ∂Γ for the boundaryof the branching random walk with offspring distribution ν and underlyingrandom walk ξ, and h(ν) for the Hausdorff dimension of Λ. When λ > 1/ρ, thebranching random walk is recurrent, trivially Λ = ∂Γ, h(ν) = dim(∂Γ).
In this talk, we focus on the transient setting i.e. λ ∈ [1,1/ρ], and prove the following results: h(ν) is adeterministic function of λ and thus denote it by h(λ); and h(λ) is continuousand strictly increasing in λ ∈ [1,1/ρ] and h(1/ρ) ≤ 1 2 dim(∂Γ); and thereis a positive constant C such that h(1/ρ) − h(λ) ∼ Cp1/ρ − λ as λ ↑ 1/ρ.
The above results confirm a conjecture of S. Lalley in his ICM 2006 Lecture:The critical exponent for Hausdorff dimensions of boundaries of branchingrandom walks on hyperbolic groups is 1/2.