Spread-out Measures in Random Walks

• Spread-out measures are probability distributions on locally compact groups whose convolution power becomes absolutely continuous with respect to Haar measure, ensuring robust convergence in random walks.
• The methodology employs techniques like Doeblin minorization, continuous time embedding, and Lyapunov functions to establish mixing rates, heat kernel bounds, and limit theorems.
• Applications span finite volume spaces and quantum walks, where spread-out distributions enable rigorous analysis of ergodicity, recurrence, and sub-ballistic spreading in disordered environments.

A measure on a locally compact group is termed spread-out if some convolution power is not singular with respect to Haar measure. Spread-out measures play a central role in the long-term behavior of random walks on both groups and homogeneous spaces, yielding fundamental results on ergodicity, return probabilities, limit theorems, mixing rates, and recurrence. Spread-out distributions also appear in the study of quantum and classical walks with non-local increments and are essential for understanding convergence to equilibrium, especially in infinite groups or spaces of polynomial growth.

1. Definition and Characterizations of Spread-Out Measures

Let $G$ denote a $\sigma$-compact, locally compact, metrizable group, and let $m_G$ be a choice of left Haar measure on $G$. A Borel probability measure $\mu$ on $G$ is spread-out if there exists $n_0\in\mathbb{N}$ such that the $n_0$-fold convolution $\mu^{*n_0}$ is absolutely continuous with respect to Haar measure: $\mu^{*n_0} \ll m_G$. This guarantees that the random walk can reach substantial regions of $\sigma$0 with non-negligible probability. In the context of discrete groups, a typical example is a symmetric measure with a tail that decays as

$\sigma$1

where $\sigma$2 denotes the word length with respect to a finite generating set and $\sigma$3 is the associated volume growth function (Saloff-Coste et al., 2013).

A measure is adapted if the closed subgroup generated by $\sigma$4 is all of $\sigma$5. Aperiodicity is defined by requiring $\sigma$6 to not be supported on a coset of any proper open normal subgroup containing the commutator subgroup $\sigma$7.

For random walks on quotient spaces $\sigma$8, a measure $\sigma$9 on $m_G$0 is spread-out if one of its convolution powers is not singular with respect to Haar measure on $m_G$1, assuming such a measure exists (Prohaska, 2019).

2. Spread-Out Measures and the Markov Chain Framework

The transition operator for the random walk on $m_G$2 driven by $m_G$3 is given by

$m_G$4

with the $m_G$5-step law from $m_G$6 given by $m_G$7. This Markov chain is called $m_G$8-irreducible if, for some $m_G$9-finite measure $G$0 on $G$1, every set of positive $G$2-measure is visited by the chain started from any $G$3 with positive probability.

Key equivalence:

• The random walk is a T-chain (admits a nontrivial continuous minorant) if and only if $G$4 is spread-out.
• For spread-out, adapted measures (or on finite volume spaces), the chain is $G$5-irreducible with maximal irreducibility measure equivalent to Haar measure $G$6 (Prohaska, 2019).

Aperiodicity of $G$7 on $G$8 implies aperiodicity on $G$9. Consequently, spread-out measures induce Markov chains with favorable ergodic properties, enabling a full analysis via Harris recurrence and positive recurrence theory.

3. Random Walks on Homogeneous Spaces with Spread-Out Measures

Finite Volume Case

For $\mu$0 a lattice and $\mu$1 admitting a finite $\mu$2-invariant measure, the following holds for adapted, aperiodic, spread-out $\mu$3:

• The induced Markov chain is positive Harris recurrent and aperiodic.
• There is a unique invariant probability measure, the normalized Haar measure on the orbit.

Equidistribution:

As $\mu$4,

$\mu$5

for all $\mu$6, with total variation convergence to Haar measure (Prohaska, 2019).

Exponential convergence:

If there exists a Foster–Lyapunov function $\mu$7 satisfying

$\mu$8

uniform convergence to equilibrium is exponentially fast on compact subsets, and uniformly in $\mu$9 for compact $G$0.

Limit theorems:

• SLLN: For $G$1, $G$2 almost surely.
• CLT: If $G$3, $G$4 converges in distribution to a normal law.
• LIL: Almost sure upper fluctuation limit given by

$G$5

(Prohaska, 2019)

Infinite Volume, Polynomial Growth

For infinite volume spaces of at most quadratic growth equipped with a symmetric, adapted, spread-out, compactly supported measure, the random walk is topologically Harris recurrent: from every point and every neighborhood, the walk returns infinitely often with probability one.

Ratio limit theorem: If the walk is Harris recurrent, then for any two starting points $G$6 and compactly supported, nonnegative bounded functions $G$7,

$G$8

With additional symmetry and aperiodicity, this extends to non-averaged limits for starting measures with bounded densities (Prohaska, 2019).

4. Analytic Techniques for Spread-Out Measures

Several technical methods underpin the study of spread-out measures:

1. Doeblin Minorization and Small Sets: Spread-out implies existence of $G$9 so that $n_0\in\mathbb{N}$0 is absolutely continuous with density bounded below on a compact region, leading to small-set conditions and contractive couplings (Prohaska, 2019).
2. Continuous Time Embedding and Dirichlet Forms: Continuous-time analogues permit application of analytic semigroup tools; for symmetric kernels, Dirichlet forms control the mixing and return probability asymptotics (Saloff-Coste et al., 2013).
3. Davies’s Method and Meyer's Tightness: Used to derive off-diagonal heat kernel bounds and control large jumps by truncating to a ball of radius $n_0\in\mathbb{N}$1.
4. Pseudo-Poincaré Inequalities and Two-Sided Bounds: Key for random walks with critical tails; they permit two-sided on-diagonal estimates indicating the sharp decay of return probabilities (Saloff-Coste et al., 2013).
5. Lyapunov Functions for Quantitative Rates: A drift inequality $n_0\in\mathbb{N}$2 yields geometric ergodicity and exponential mixing (Prohaska, 2019).
6. Orey–Nummelin Techniques: Enable derivation of ratio limit results under symmetry, Harris recurrence, and minorization conditions.

5. Spread-Out Measures in Groups of Polynomial Volume Growth

In finitely generated groups $n_0\in\mathbb{N}$3 of polynomial volume growth $n_0\in\mathbb{N}$4, spread-out measures with critical tail, such as

$n_0\in\mathbb{N}$5

satisfy sharp asymptotics for the return probability:

$n_0\in\mathbb{N}$6

Key features of the proof include controlling the kernel by truncation, log-Sobolev inequalities, Gaussian-type off-diagonal bounds, and strong two-sided control via pseudo-Poincaré inequalities (Saloff-Coste et al., 2013).

Extensions capture regularly varying perturbations, stable-like tails, and measures supported on powers of generators in nilpotent groups. For example, for stable-like tails $n_0\in\mathbb{N}$7, the return probability decays as $n_0\in\mathbb{N}$8 (Saloff-Coste et al., 2013).

6. Quantum Random Walks and Spread-Out Step Distributions

In the context of quantum walks, spread-out measures are realized via randomized step-lengths, such as quenched Poisson-distributed disorder:

• In the ordered (homogeneous) quantum walk, interference produces ballistic spreading: variance grows as $n_0\in\mathbb{N}$9.
• When step-lengths are i.i.d. and Poisson-distributed, the spread of the walker is inhibited: after disorder-averaging, the variance grows as $n_0$0, corresponding to $n_0$1 (Das et al., 2018).

This scaling is sub-ballistic yet super-diffusive, distinct from the standard ballistic ($n_0$2) and classical diffusive ($n_0$3) regimes. The observed exponent is universal for a large class of sub- and super-Poissonian jump distributions, provided the law has finite mean and variance. Thus, introducing spread-out randomness into quantum walks consistently slows but does not fully localize the spread, retaining a quantum advantage over classical random walks (Das et al., 2018).

7. Consequences and Applications

Spread-out measures underpin key phenomena in ergodic theory, probability on groups, and quantum walks:

• Mixing times: For spread-out random walks, convergence in total variation to equilibrium is typically of order $n_0$4, up to logarithmic factors, with precise behavior governed by tail regularity and group growth.
• Heat kernel decay: With critical tail measures, the decay matches that of balls of radius $n_0$5.
• Recurrence criteria: Spread-out, symmetric, adapted measures on spaces of at most quadratic growth yield Harris recurrence.
• Limit theorems: Classical results (SLLN, CLT, LIL) extend under minimal tail assumptions due to spread-outness.
• Quantum-classical contrast: Quantum walks exhibit persistent faster-than-diffusive spread even under spread-out measure-induced disorder.

These properties have deep implications in the study of random walks on groups, homogeneous spaces, and quantum systems, and remain active areas for further research (Prohaska, 2019, Saloff-Coste et al., 2013, Das et al., 2018).