Shannon’s theorem for locally compact groups

Stationary random subgroups in negative curvature

“…In the focal case, the extended set of assumptions implies that the Poisson boundary of the pair (G, µ) can be identified with the Gromov boundary ∂X equipped with the µ-stationary hitting measure ν X , see [FT22,Theorem 1.4]. See [Kai96] for a closely related survey of the identification problem for Poisson boundaries.…”

Strong Shannon–McMillan–Breiman’s theorem for locally compact groups

“…Derriennic [Der80] asked if one can establish similar results in the context of random walks on noncountable locally compact groups. Although the analog of the Shannon–McMillan–Breiman theorem for stationary processes on continuous spaces was completed in 80s, its analog for random walks on noncountable locally compact groups remained unsolved in the last 40 years, until, in a recent result [FT22], Forghani and Tiozzo provided a weak version of the Shannon–McMillan–Breiman theorem for random walks on locally compact groups.…”

“…For instance, when the asymptotic entropy is finite, if and only if all bounded harmonic functions are constant (equivalently, the Poisson boundary is trivial; see [Der80, KV]. A version of the Shannon–McMillan–Breiman theorem is used to prove ray and strip approximations , fundamental criteria to identify the Poisson boundary and bounded harmonic functions of a random walk (see [FT22, Kai00] for more details).…”

“…Conjecture 1.1 has been solved when G is a countable group by Kaimanovich and Vershik [KV] and Derriennic [Der80] by using the subadditive ergodic theorem. In the recent development, Forghani and Tiozzo [FT22] established a weak version of Conjecture 1.1 for random walks on locally compact groups, that is, for -almost every sample path , …”

Compactifications of horospheric products

“…Ever since their introduction, these graphs have been a source of many challenging questions that involve an interplay between probabilistic and geometrical aspects. Some of these work include convergence of random walks [Ber01], spectral radius of simple random walk [SCW97] and [BW05a], Martin boundary and minimal harmonic functions in [BW05b] and [BW06], the Poisson boundaries of discrete isometry groups of horospheric products [BNW08] [KS12] and the Poisson boundaries of locally compact isometry groups of horospheric products [FT22]. It is also worth pointing out that, disparate as they might appear, the Martin compactification, the Busemann compactification, and the Thurston compactification of the Teichmüller space have the common feature that they are all based on embedding a certain space X into a projective space of functions on X, and then closing it.…”