---
title: Spread-out Measures in Random Walks
url: https://www.emergentmind.com/topics/spread-out-measures-in-random-walks
type: topic
---

# Spread-out Measures in Random Walks

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 $G$ with non-negligible probability. In the context of discrete groups, a typical example is a symmetric measure with a tail that decays as
$$
\nu(g) \asymp \frac{1}{(1+|g|)^2 V(|g|)},
$$
where $|g|$ denotes the word length with respect to a finite generating set and $V(r) = \#\{g \in G : |g|\leq r\}$ is the associated volume growth function [1309.6296].

A measure is **adapted** if the closed subgroup generated by $\mathrm{supp}(\mu)$ is all of $G$. **Aperiodicity** is defined by requiring $\mu$ to not be supported on a coset of any proper open normal subgroup containing the commutator subgroup $[G,G]$.

For random walks on quotient spaces $X = G/\Gamma$, a measure $\mu$ on $G$ is spread-out if one of its convolution powers is not singular with respect to Haar measure on $X$, assuming such a measure exists [1910.00467].

## 2. Spread-Out Measures and the Markov Chain Framework

The transition operator for the random walk on $X=G/\Gamma$ driven by $\mu$ is given by
$$
P(x,A) = \mu * \delta_x(A),
$$
with the $n$-step law from $x$ given by $\mu^{*n}*\delta_x$. This Markov chain is called **$\psi$-irreducible** if, for some $\sigma$-finite measure $\psi$ on $X$, every set of positive $\psi$-measure is visited by the chain started from any $x \in X$ with positive probability.

Key equivalence:  
- The random walk is a **T-chain** (admits a nontrivial continuous minorant) if and only if $\mu$ is spread-out.
- For spread-out, adapted measures (or on finite volume spaces), the chain is $\psi$-irreducible with maximal irreducibility measure equivalent to Haar measure $[m_X]$ [1910.00467].

Aperiodicity of $\mu$ on $G$ implies aperiodicity on $X$. 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 $\Gamma < G$ a lattice and $X = G/\Gamma$ admitting a finite $G$-invariant measure, the following holds for adapted, aperiodic, spread-out $\mu$:
- 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 $n\to\infty$,
$$
\| \mu^{*n}*\delta_{x_0} - m_{Gx_0} \|_{TV} \to 0,
$$
for all $x_0 \in X$, with total variation convergence to Haar measure [1910.00467].

**Exponential convergence:**  
If there exists a Foster–Lyapunov function $V:X \to [1,\infty)$ satisfying 
$$
PV(x)\leq\alpha V(x) + \beta, \quad 0<\alpha<1, \beta<\infty,
$$
uniform convergence to equilibrium is exponentially fast on compact subsets, and uniformly in $x$ for compact $X$.

**Limit theorems:**  
- **SLLN:** For $f \in L^1(m_X)$, $(1/n)\sum_{k=0}^{n-1} f(\Phi_k) \to \int_X f\, dm_X$ almost surely.
- **CLT:** If $f \in L^2(m_X)$, $(1/\sqrt{n})\sum_{k=0}^{n-1} (f(\Phi_k) - E[f])$ converges in distribution to a normal law.
- **LIL:** Almost sure upper fluctuation limit given by
$$
\limsup_{n\to\infty}\frac{\sum_{k=0}^{n-1}(f(\Phi_k)-E[f])}{\sqrt{2n\log\log n}} = \sigma_f.
$$
[1910.00467]

### 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 $x_1,x_2\in X$ and compactly supported, nonnegative bounded functions $f_1, f_2$,
$$
\frac{\sum_{j=0}^n E_{x_1}[f_1(\Phi_j)]}{\sum_{j=0}^n E_{x_2}[f_2(\Phi_j)]} \to \frac{\int f_1 dm_X}{\int f_2 dm_X}.
$$
With additional symmetry and aperiodicity, this extends to non-averaged limits for starting measures with bounded densities [1910.00467].

## 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 $n_0$ so that $\mu^{*n_0}$ is absolutely continuous with density bounded below on a compact region, leading to small-set conditions and contractive couplings [1910.00467].

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 [1309.6296].

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 $R(t) \sim \sqrt{t\log t}$.

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 [1309.6296].

5. **Lyapunov Functions for Quantitative Rates:**  
A drift inequality $PV \leq \alpha V + \beta$ yields geometric ergodicity and exponential mixing [1910.00467].

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 $G$ of polynomial volume growth $V(r) \asymp r^d$, spread-out measures with critical tail, such as
$$
\nu(g) \asymp \frac{1}{(1+|g|)^2 V(|g|)},
$$
satisfy sharp asymptotics for the return probability:
$$
\nu^{(n)}(e) \asymp \frac{1}{V(\sqrt{n \log n})} \asymp (n\log n)^{-d/2}.
$$
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 [1309.6296].

Extensions capture regularly varying perturbations, stable-like tails, and measures supported on powers of generators in nilpotent groups. For example, for stable-like tails $\nu_a(g)\asymp(1+|g|)^{-d-a}$, the return probability decays as $n^{-d/a}$ [1309.6296].

## 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 $t^2$.
- When step-lengths are i.i.d. and Poisson-distributed, the spread of the walker is **inhibited**: after disorder-averaging, the variance grows as $t^{1.6}$, corresponding to $\langle\sigma_{\text{dis}}(t)\rangle \sim t^{0.8}$ [1806.04024].

This scaling is **sub-ballistic yet super-diffusive**, distinct from the standard ballistic ($t^2$) and classical diffusive ($t$) 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 [1806.04024].

## 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 $n$, 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 $\sim\sqrt{n \log n}$.
- **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 [1910.00467, 1309.6296, 1806.04024].

Source: https://www.emergentmind.com/topics/spread-out-measures-in-random-walks