We study random walks on the giant component of Hyperbolic Random Graphs (HRGs), in the regime when the degree distribution obeys a power law with exponent in the range $(2,3)$. In particular, we first focus on the expected time for a random walk to hit a given vertex or visit, i.e. cover, all vertices. We show that, a.a.s. (with respect to the HRG), and up to multiplicative constants: the cover time is $n(\log n)^2$, the maximum hitting time is $n\log n$, and the average hitting time is $n$. We then determine the expected time to commute between two given vertices a.a.s., up to a small factor polylogarithmic in $n$, and under some mild hypothesis on the pair of vertices involved. Our results are proved by controlling effective resistances using the energy dissipated by carefully designed network flows associated to a tiling of the hyperbolic plane, on which we overlay a forest-like structure.

翻译：我们研究双曲随机图（HRGs）巨分支上的随机游走，考虑其度分布服从幂律且指数在$(2,3)$范围内的情形。具体而言，我们首先聚焦于随机游走击中给定顶点或覆盖所有顶点的期望时间。结果表明，关于HRG图的渐近几乎必然（a.a.s.）性质，在乘法常数范围内：覆盖时间为$n(\log n)^2$，最大击中时间为$n\log n$，平均击中时间为$n$。随后，我们确定了任意两个给定顶点之间通勤时间的渐近期望值（a.a.s.），该结果在$n$的多对数因子范围内成立，且需满足所涉顶点对的某些温和假设。通过利用精心设计的网络流（与双曲平面的铺砌相关联，并在其上叠加森林状结构）所耗散的能量来控制有效电阻，我们完成了上述结论的证明。