Let $\Gamma$ be a relatively hyperbolic group and let $\mu$ be an admissible symmetric finitely supported probability measure on $\Gamma$. We extend Floyd-Ancona type inequalities up to the spectral radius of $\mu$. We then show that when the parabolic subgroups are virtually abelian, the Martin boundary of the induced random walk on $\Gamma$ is stable in the sense of Picardello and Woess. We also define a notion of spectral degenerescence along parabolic subgroups and give a criterion for strong stability of the Martin boundary in terms of spectral degenerescence. We prove that this criterion is always satisfied in small rank. so that in particular, the Martin boundary of an admissible symmetric finitely supported probability measure on a geometrically finite Kleinian group of dimension at most 5 is always strongly stable.

中文翻译：

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相对双曲群马丁边界的稳定性现象

令$\Gamma$ 是一个相对双曲群，并令$\mu$ 是$\Gamma$ 上的一个可接受对称有限支持概率测度。我们将 Floyd-Ancona 类型的不等式扩展到 $\mu$ 的谱半径。然后我们证明，当抛物线子群实际上是阿贝尔群时，$\Gamma$ 上的诱导随机游走的 Martin 边界在 Picardello 和 Woess 的意义上是稳定的。我们还定义了沿抛物线子群的光谱简并的概念，并根据光谱简并给出了马丁边界强稳定性的标准。我们证明这个标准总是在小秩上满足。因此，特别是，在维度最多为 5 的几何有限克莱因群上的可容许对称有限支持概率测度的马丁边界始终是强稳定的。