Uniform spectral gaps, non-abelian Littlewood-Offord and anti-concentration for random walks

We show that random walks on semisimple algebraic groups do not concentrate on proper algebraic subvarieties with uniform exponential rate of anti-concentration. This is achieved by proving a uniform spectral gap for quasi-regular representations of countable linear groups. The method makes key use of Diophantine heights and the Height Gap theorem. We also deduce a non-abelian version of the Littlewood–Offord inequalities and prove logarithmic bounds for escape from subvarieties. In a sequel to this paper, we will show how to transform this uniform gap into uniform expansion for Cayley graphs of finite simple groups of bounded rank $G(p)$ over almost all primes $p$.

💡 Research Summary

The paper establishes a uniform anti‑concentration phenomenon for random walks on semisimple algebraic groups, together with a uniform spectral gap for a broad class of quasi‑regular representations. The authors consider a countable linear group Γ ⊂ GL_d(K) that is Zariski‑dense in a connected semisimple algebraic K‑group G. For any subgroup H ≤ Γ that is not Zariski‑dense in G, they prove that there exists a constant ε_d > 0 depending only on the dimension d such that for every finite generating set S ⊂ Γ whose image generates a Zariski‑dense subgroup of G, one has

 max_{s∈S} ‖λ_{Γ/H}(s)v – v‖ ≥ ε_d

for every unit vector v in ℓ²(Γ/H). This is a uniform spectral gap: the Kazhdan constant ε_d does not depend on the field K, its characteristic, nor on the particular subgroup H, as long as H is not Zariski‑dense. The proof hinges on a new “ping‑pong with overlaps” lemma, Diophantine height arguments, and the Height Gap theorem of Breuillard–Gamburd–Saxcé, which together give an effective lower bound for ε_d.

From the uniform gap the authors deduce a quantitative bound on the operator norm of any probability measure μ on Γ:

 ‖λ_{Γ/H}(μ)‖_{op} ≤ 1 – β(μ)·ε’_d, ε’_d = ε_d²/2,

where

 β(μ) = 1 – sup_{L} μ(gL ∩ Γ)

and the supremum runs over all proper algebraic subgroups L of G. Thus, unless μ is almost entirely supported on a coset of a proper algebraic subgroup, the associated averaging operator contracts uniformly.

The main probabilistic application is an anti‑concentration inequality for products of independent G‑valued random variables X₁,…,X_n. Define

 β(X_i) = 1 – sup_{g∈G, L≤G proper} P(X_i ∈ gL).

Then for any proper algebraic subvariety V ⊂ G,

 P(X₁·…·X_n ∈ V) ≤ C_V·exp(–c·∑_{i=1}^n β(X_i)),

where C_V = (1+dim V)^{1/2}·deg(V) (C_V = 1 when V is a coset of a subgroup) and c = ε’_d · 4^{–d²}. This bound is uniform: it depends on V only through its degree and dimension, and on the random variables only through the β‑parameters, which measure how far each variable is from being concentrated on a proper algebraic subgroup.

A striking corollary is obtained when each X_i is uniformly distributed on a finite set S_i of size k, and the set S_i^{-1}S_i generates a Zariski‑dense subgroup of G. In this case β(X_i) ≥ 1/k, and the inequality simplifies to

 P(X₁·…·X_n ∈ V) ≤ C_V·exp(–c·n/k),

showing exponential decay with a rate that depends only on d and k, not on the specific distributions or the geometry of the sets S_i.

When V is a single point, the estimate becomes a non‑abelian Littlewood–Offord inequality:

 sup_{g∈GL_d(k)} P(X₁·…·X_n = g) ≤ exp(–c_d·∑_{i=1}^n β_i),

with β_i defined analogously using virtually solvable subgroups. This improves on earlier results that gave only inverse‑square‑root bounds; here the decay is exponential provided the variables are not overly supported on virtually solvable subgroups.

The authors also prove a logarithmic escape lemma: if a finite symmetric generating set S ⊂ G(K) is Zariski‑dense and Sⁿ is contained in a proper algebraic subvariety V of degree ≤ N, then

 n ≤ C_d·log(1+N).

This strengthens earlier polynomial‑degree bounds (e.g., those obtained via Bézout) and follows from the uniform anti‑concentration estimates.

The paper discusses several auxiliary topics: weak containment of quasi‑regular representations, connections with invariant means and paradoxical decompositions, alternative proofs of spectral gaps without uniformity (via Furstenberg’s Borel density argument), and the relationship with super‑strong approximation. The authors emphasize that the uniform spectral gap for λ_{Γ/H} cannot be deduced from the regular representation when H is non‑amenable, highlighting the novelty of their approach.

Finally, the authors outline forthcoming work where the uniform gap is transferred to uniform expansion for Cayley graphs of finite simple groups G(p) of bounded rank over almost all primes p, and where these expansion results are applied to the study of character varieties of random groups.

In summary, the paper delivers a powerful, uniform spectral gap for quasi‑regular representations, translates it into exponential anti‑concentration for random walks on semisimple groups, provides a strong non‑abelian Littlewood–Offord bound, and yields logarithmic escape estimates—all with explicit dependence only on the ambient dimension and elementary parameters of the random variables. This unifies and significantly strengthens several strands of research at the intersection of representation theory, algebraic geometry, and probabilistic group theory.