Multislicing and effective equidistribution for random walks on some homogeneous spaces

We consider a random walk on a homogeneous space $G/Λ$ where $G$ is $\mathrm{SO}(2,1)$ or $\mathrm{SO}(3,1)$ and $Λ$ is a lattice. The walk is driven by a probability measure $μ$ on $G$ whose support generates a Zariski-dense subgroup. We show that for every starting point $x \in G/Λ$ which is not trapped in a finite $μ$-invariant set, the $n$-step distribution $μ^{*n}*δ_{x}$ of the walk equidistributes toward the Haar measure. Moreover, under arithmetic assumptions on the pair $(Λ, μ)$, we show the convergence occurs at an exponential rate, tempered by the obstructions that $x$ may be high in a cusp or close to a finite orbit. Our approach is substantially different from that of Benoist-Quint, whose equidistribution statements only hold in Cesàro average and are not quantitative, that of Bourgain-Furman-Lindenstrauss-Mozes concerning the torus case, and that of Lindenstrauss-Mohammadi-Wang and Yang about the analogous problem for unipotent flows. A key new feature of our proof is the use of a new phenomenon which we call multislicing. The latter is a generalization of the discretized projection theorems à la Bourgain and we believe it presents independent interest.

💡 Research Summary

The paper studies random walks on homogeneous spaces X = G/Λ where G is a real linear group locally isomorphic to SO(2,1) or SO(3,1) and Λ is a lattice. A probability measure μ on G drives the walk; μ is assumed to have a finite exponential moment and its support generates a Zariski‑dense subgroup Γ_μ of G. The authors prove three main results.

First, Theorem 1.1 establishes that for any starting point x∈X the n‑step distribution μ^{*n}δ_x converges weak‑ to the G‑invariant Haar probability measure m_X, provided the Γ_μ‑orbit of x is infinite. This removes the Cesàro‑average hypothesis present in the earlier work of Benoist–Quint and gives a direct convergence statement. Consequently, the paper recovers the rigidity statements of Benoist–Quint (classification of stationary measures, orbit closures, and stiffness) via a completely different method that does not rely on their earlier arguments.

Second, Theorem 1.2 provides a quantitative version. If the initial distribution ν on X satisfies a Frostman‑type condition ν(B_ρ(x)) ≤ ρ^κ for all scales ρ between δ and δ^ε (with κ>0 fixed), then after n ≥ |log δ| steps the walk equidistributes with an explicit error bound |μ^{*n}*ν(f) – m_X(f)| ≤ δ^ε + ν{inj ≤ δ^ε} for any Lipschitz test function f with ‖f‖_Lip ≤ 1. Here inj(x) denotes the injectivity radius, measuring how deep x lies in a cusp. This theorem shows that once the walk acquires positive dimension at some scale, exponential convergence follows.

Third, Theorem 1.3 treats the case of arithmetic lattices and algebraic measures. Assume Λ is arithmetic and μ is algebraic with respect to Λ (i.e., all matrix entries of Ad(Γ_μ) and Ad(Λ) are algebraic numbers). Then for any x∈X, any R≥2 and any Lipschitz f, the error |μ^{*n}*δ_x(f) – m_X(f)| is bounded by R^{–1}‖f‖Lip as soon as
n ≥ A·log R + A·max{ |log d(x, W{μ,R})| , d(x, x_0) } .
Here x_0 is the base point Λ∈X, W_{μ,R} is the set of points whose Γ_μ‑orbit has size ≤R, and d denotes the ambient metric. The term involving d(x, W_{μ,R}) reflects the time needed for the walk to escape a small neighbourhood of a finite orbit, while the term d(x, x_0) measures the time needed to leave a deep cusp. The constant A depends only on G, Λ and μ.

The novelty of the paper lies in the method. Classical approaches to effective equidistribution on homogeneous spaces (e.g., Green–Tao for nilmanifolds, Bourgain–Furman–Lindenstrauss–Mozes for the torus) rely heavily on Fourier analysis and spectral gap for linear actions. In the non‑commutative setting of SO(2,1) and SO(3,1) such tools are unavailable. The authors introduce a new combinatorial‑geometric device called “multislicing,” which generalizes Bourgain’s discretized projection theorems to a setting where one simultaneously slices along several directions. This allows them to increase the dimension of a measure in many directions at once, even when the measure is not Lebesgue (e.g., a Furstenberg measure of positive but non‑full Hausdorff dimension).

The proof proceeds in three stages. First, using the multislicing theorem, they show that after a bounded number of steps the random walk pushes any initial distribution to one that has positive dimension κ at some scale. Second, they iterate the walk and a “dimension boot‑strap” argument to raise the dimension arbitrarily close to the full dimension of X (which is 3 for SO(2,1) and 6 for SO(3,1)). This step requires careful control of covering numbers and uses sub‑ and super‑critical estimates for covering numbers of slices. Third, once the measure has high dimension, a spectral gap argument for the Markov operator associated to μ yields exponential convergence to the Haar measure.

The paper also contains a detailed analysis of the arithmetic case. Algebraicity of μ and Λ allows the authors to control Diophantine properties of the orbit, to bound the size of finite Γ_μ‑orbits, and to obtain effective recurrence estimates. These ingredients are crucial for turning the qualitative equidistribution of Theorem 1.1 into the quantitative bound of Theorem 1.3.

In addition to the main results, the authors prove auxiliary statements: an effective positive dimension theorem (Section 3), a robust measure framework (Section 4), and a non‑linear sub‑critical projection theorem (Appendix A). They also discuss connections to prior work on unipotent flows (Lindenstrauss–Mohammadi–Wang, Yang) and highlight how the lack of a distinguished unstable direction in the random walk setting forces a genuinely higher‑dimensional slicing approach.

Overall, the paper delivers the first effective equidistribution theorem for random walks on non‑nilpotent, rank‑one homogeneous spaces without averaging, and introduces multislicing as a versatile tool that may find applications beyond the present setting, such as in Diophantine approximation on fractals and in the study of stationary measures for other non‑commutative actions.