Quenched Central Limit Theorem

• Quenched central limit theorem is a Gaussian limit theorem conditioned on a fixed random environment, ensuring convergence with deterministic covariance.
• It employs methods such as martingale reductions, corrector techniques, and regeneration strategies to control environment-specific drift.
• Its applications span random walks, stationary processes, and Fourier transforms, highlighting the distinction between quenched and annealed convergence.

A quenched central limit theorem is a Gaussian limit theorem proved after freezing an external source of randomness—typically a random environment, an initial past, a random graph, or a random dynamical system—and then taking the limit under the corresponding conditional law for almost every realization. In this sense, quenched convergence is stronger than annealed convergence: it is a statement about typical fixed environments or typical starting conditions rather than about averages over them. Across random walks in random media, stationary processes, Fourier transforms, random fields, polymers, and random dynamical systems, the recurring structure is a centered and normalized observable whose conditional law converges to a Gaussian or Brownian limit with deterministic covariance, but the correct centering, variance identification, and proof strategy are model-dependent (Peligrad, 2022, Tóth, 2017, Barrera et al., 2014).

1. Meaning of “quenched” and standard formulations

The basic quenched formulation is conditional weak convergence. For a stationary process generated on an ergodic dynamical system, one fixes a regular conditional probability $m_\omega$ with respect to a past $\sigma$-algebra and asks whether, for $\mu$-almost every $\omega$,

$$m_\omega\!\left(\frac{S_n}{a_n}\in\cdot\right)\Rightarrow \mathcal N(0,\sigma^2).$$

In the linear-process setting this is explicitly interpreted as a CLT “started from almost every initial environment/past,” and for stationary Markov chains it is equivalent to the started-at-a-point formulation $P^x(S_n/\sqrt n\le t)\to \Phi_{\sigma^2}(t)$ for $\pi$-almost every $x$ (Volny et al., 2015, Peligrad, 2022).

For random walks in random environment, the quenched law is usually denoted $P_\omega$ or $P^\omega$: the environment $\sigma$0 is fixed first, and the walk is then studied conditionally on that realization. In the doubly stochastic setting, this means that for $\sigma$1-almost every $\sigma$2, the diffusive scaling $\sigma$3 converges in law under $\sigma$4 to Brownian motion with deterministic covariance; in the time-dependent balanced setting, the same philosophy yields not only a quenched invariance principle but also a quenched local central limit theorem for the heat kernel (Tóth, 2017, Deuschel et al., 2017).

The same terminology extends beyond classical random media. For the discrete Fourier transform of a stationary ergodic process, quenched means conditioning on the past $\sigma$5; for T-graphs it means fixing a typical value of the quasi-periodic parameter $\sigma$6 and studying the walk on the resulting deterministic graph; for the Ising model on random graphs it means freezing the graph $\sigma$7 and then sampling spins from the Gibbs measure on that graph (Barrera et al., 2014, Laslier, 2013, Giardina' et al., 2014).

Setting	Frozen object	Typical quenched statement
Stationary process	past $\sigma$8-algebra	conditional law converges a.s.
Markov chain	starting point $\sigma$9	CLT for $\mu$0-a.e. $\mu$1
RWRE	environment $\mu$2	$\mu$3-CLT a.s. in $\mu$4
Random graph model	graph $\mu$5	CLT under $\mu$6
Random dynamical system	base path $\mu$7	fibrewise Gaussian limit

A central conceptual point is that quenched convergence is not a cosmetic strengthening of annealed convergence. Several papers emphasize that annealed CLT can hold while quenched CLT fails, and that even when both hold, the required centering can be different (Volny et al., 2015, Volný et al., 2010).

2. Core mechanisms and proof architectures

One dominant mechanism is martingale reduction. In the random walk model introduced by Boldrighini, Minlos, and Pellegrinotti, the walk on $\mu$8 has one-step kernel $\mu$9, with i.i.d. space-time environment $\omega$0, bounded range, and zero-mean perturbation under the environment law. Writing

$\omega$1

the centered process $\omega$2, with $\omega$3, is a martingale under the quenched law for every fixed environment. The proof then identifies the limiting covariance

$\omega$4

through convergence of quadratic variation and applies the multidimensional martingale CLT of Kückler–Sørensen (Bezborodov et al., 2017).

A second canonical architecture is the corrector method. For nearest-neighbor walks in doubly stochastic random environments, the drift is decomposed into symmetric and antisymmetric parts, the $\omega$5-condition yields a square-integrable stream-tensor representation, and a harmonic cocycle $\omega$6 is constructed so that the corrected displacement $\omega$7 is a quenched martingale. The remaining task is to prove sublinearity of the corrector at diffusive scale and tightness of the walk. In that setting the proof combines an extension of Nash’s moment method to divergence-free drift, functional-analytic construction of harmonic coordinates, and a martingale CLT (Tóth, 2017).

Regeneration is the other major paradigm. For biased random walks on supercritical Galton–Watson trees with leaves, the walk is decomposed into i.i.d. centered regeneration increments once one works along the backbone and proves finite second moments of trap-return times. The quenched upgrade then follows through a Bolthausen–Sznitman-type variance argument comparing two walks on the same tree. For ancestral lineages in logistic branching random walks, regeneration times are built through coarse graining, cone-shell shielding, and coupling of two walks in the same medium to walks in independent media (Bowditch, 2017, Birkner et al., 2024).

Spectral and mixing methods form a separate line. For expanding-on-average cocycles, the proof adapts the non-autonomous spectral method of Dragičević–Froyland–Gonzalez-Tokman–Vaienti to Buzzi’s nonuniform setting by introducing adapted norms based on the Oseledets splitting, analytic perturbation of twisted transfer operators, and a Green–Kubo variance formula. In the more abstract framework of random dynamical systems with quenched mixing of all orders, the proof is a method-of-moments argument in which only paired correlations survive, with random thresholds $\omega$8 replacing uniform constants in the mixing estimates (Dragičević et al., 2021, Auer, 29 May 2026).

Concentration methods also appear in genuinely non-Markovian models. In the corner-growth setting, the quenched CLT for the energy of a typical up-right path is derived from a concentration lemma for convex 1-Lipschitz test functions, Talagrand-type concentration for truncated environments, and combinatorial control of path overlaps through the quantity

$\omega$9

There the overlap exponent $$m_\omega\!\left(\frac{S_n}{a_n}\in\cdot\right)\Rightarrow \mathcal N(0,\sigma^2).$$0 determines the moment condition needed for the environment (Gromoll et al., 2018).

3. Random walks in random and dynamic media

The random-walk literature provides the classical template for quenched Gaussian limits. In the i.i.d. dynamical environment of Boldrighini–Minlos–Pellegrinotti, the environment is a finite-valued i.i.d. space-time field, the one-step law is $$m_\omega\!\left(\frac{S_n}{a_n}\in\cdot\right)\Rightarrow \mathcal N(0,\sigma^2).$$1, and bounded range plus drift cancellation imply that $$m_\omega\!\left(\frac{S_n}{a_n}\in\cdot\right)\Rightarrow \mathcal N(0,\sigma^2).$$2 is a quenched martingale. The theorem identifies a deterministic covariance matrix $$m_\omega\!\left(\frac{S_n}{a_n}\in\cdot\right)\Rightarrow \mathcal N(0,\sigma^2).$$3 from the averaged jump distribution and yields a diffusive Gaussian limit for $$m_\omega\!\left(\frac{S_n}{a_n}\in\cdot\right)\Rightarrow \mathcal N(0,\sigma^2).$$4-almost every environment (Bezborodov et al., 2017).

A substantially more delicate case is the doubly stochastic random environment with divergence-free drift. Here the walk is continuous-time, nearest-neighbor, and driven by bounded rates $$m_\omega\!\left(\frac{S_n}{a_n}\in\cdot\right)\Rightarrow \mathcal N(0,\sigma^2).$$5 satisfying double stochasticity and uniform ellipticity of the symmetric part. Under the $$m_\omega\!\left(\frac{S_n}{a_n}\in\cdot\right)\Rightarrow \mathcal N(0,\sigma^2).$$6-condition and an additional $$m_\omega\!\left(\frac{S_n}{a_n}\in\cdot\right)\Rightarrow \mathcal N(0,\sigma^2).$$7 bound on the stream tensor, the quenched invariance principle holds for $$m_\omega\!\left(\frac{S_n}{a_n}\in\cdot\right)\Rightarrow \mathcal N(0,\sigma^2).$$8, with nondegenerate covariance given through the harmonic-coordinate construction. The result strengthens earlier convergence in probability with respect to the environment to almost sure quenched convergence (Tóth, 2017).

One-dimensional RWRE shows that quenched CLT may require random centering and exhibits nonclassical rates. For i.i.d. environments with

$$m_\omega\!\left(\frac{S_n}{a_n}\in\cdot\right)\Rightarrow \mathcal N(0,\sigma^2).$$9

the quenched CLT for hitting times is standard after centering by $P^x(S_n/\sqrt n\le t)\to \Phi_{\sigma^2}(t)$0, while the position CLT requires the environment-dependent correction

$P^x(S_n/\sqrt n\le t)\to \Phi_{\sigma^2}(t)$1

Ahn and Peterson obtained polynomial almost-sure Berry–Esseen-type bounds whose exponents depend on $P^x(S_n/\sqrt n\le t)\to \Phi_{\sigma^2}(t)$2, and later work proved that these upper bounds are optimal for hitting times: for $P^x(S_n/\sqrt n\le t)\to \Phi_{\sigma^2}(t)$3 the normalized error oscillates between $P^x(S_n/\sqrt n\le t)\to \Phi_{\sigma^2}(t)$4 and $P^x(S_n/\sqrt n\le t)\to \Phi_{\sigma^2}(t)$5, while at $P^x(S_n/\sqrt n\le t)\to \Phi_{\sigma^2}(t)$6 the optimal scale is $P^x(S_n/\sqrt n\le t)\to \Phi_{\sigma^2}(t)$7 (Ahn et al., 2017, Ahn et al., 2020).

The quenched program extends to more structured geometries. On quasi-periodic T-graphs, the continuous-time walk is balanced and hence a martingale. For generic $P^x(S_n/\sqrt n\le t)\to \Phi_{\sigma^2}(t)$8, the rescaled walk converges to Brownian motion with deterministic covariance $P^x(S_n/\sqrt n\le t)\to \Phi_{\sigma^2}(t)$9, and discrete harmonic functions from the dimer correspondence force $\pi$0, even though the graph has no obvious rotational symmetry. On supercritical Galton–Watson trees with leaves, a biased walk has a quenched functional CLT in the positive-speed regime

$\pi$1

with the upper bias threshold identified as sharp in the sense described in the paper and linked to a conjecture of Ben Arous and Fribergh (Laslier, 2013, Bowditch, 2017).

Dynamic random environments generated by interacting particle systems provide another class. For ancestral lineages in logistic branching random walks, the lineage evolves backward in the time-reversed population field with transition kernel

$\pi$2

In the high-density regime covered by Assumption 2.1, the paper proves that $\pi$3 converges quenchedly to a centered nondegenerate Gaussian law, using coarse graining to an oriented-percolation-type model and a regeneration construction for two walks in the same medium (Birkner et al., 2024).

A local version of the theory appears for balanced time-dependent environments on $\pi$4. Under uniform ellipticity and ergodicity under space-time shifts, one obtains not merely path-level convergence but a quenched local central limit theorem for the normalized kernel

$\pi$5

with

$\pi$6

uniformly on compact macroscopic space-time regions away from $\pi$7. This sharpens the quenched invariance principle to pointwise heat-kernel asymptotics (Deuschel et al., 2017).

4. Stationary processes, Markov chains, Fourier transforms, and random fields

In the ergodic-theoretic setting, a fundamental issue is whether annealed projective conditions are strong enough for conditional Gaussian limits. One positive result is that Hannan’s condition,

$\pi$8

implies a quenched CLT for the centered sums

$\pi$9

whereas an $x$0 coboundary decomposition does not guarantee quenched convergence for the uncentered sums. The counterexample shows that the conditional expectation can remain of order $x$1, so a telescoping representation alone is insufficient (Volný et al., 2010).

This sensitivity persists for stationary linear processes. For causal linear processes $x$2, a sufficient coefficient condition is

$x$3

which ensures a quenched CLT for

$x$4

But the same paper constructs a process satisfying both Maxwell–Woodroofe and Hannan conditions, with $x$5, for which the ordinary CLT under normalization by $x$6 holds while the quenched CLT fails and the weak invariance principle fails. This sharply separates annealed CLT, quenched CLT, and WIP in the variance-normalized regime (Volny et al., 2015).

For stationary Markov chains, the paper on started-at-a-point quenched CLT introduces a two-sided conditioning device. Besides the Cesàro boundedness condition

$x$7

it assumes

$x$8

This yields an annealed CLT and, combined with convergence of conditional second moments, characterizes quenched CLT. A new sufficient projective criterion is

$x$9

which is stronger than the insufficient $P_\omega$0-summability of $P_\omega$1 and is designed specifically for pointwise conditioning (Peligrad, 2022).

The Fourier-transform setting gives a frequency-wise variant of the theory. For a stationary ergodic process, the centered discrete Fourier transform

$P_\omega$2

satisfies a quenched CLT for almost all $P_\omega$3, with asymptotically independent Gaussian real and imaginary parts. The uncentered quenched CLT holds for almost all $P_\omega$4 if and only if

$P_\omega$5

Under regularity, the variance is $P_\omega$6, so the corresponding periodogram converges quenchedly to $P_\omega$7 (Barrera et al., 2014).

Multi-parameter analogues occur for stationary random fields. Under a commuting filtration and a Hannan-type Orlicz condition

$P_\omega$8

one obtains a quenched functional CLT for the randomly centered sums $P_\omega$9, with limit a Brownian sheet. The paper emphasizes that this weakens earlier $P^\omega$0 assumptions to the borderline Orlicz framework and extends the ortho-martingale theory of Peligrad and Volný to Hannan-type random fields (Reding et al., 2021).

5. Random dynamical systems and other nonclassical settings

Random dynamical systems furnish a broad setting in which quenched CLT is proved fibrewise. For expanding-on-average cocycles over an ergodic base, the transfer-operator cocycle is not uniformly expanding in $P^\omega$1, and the observable must be renormalized by a tempered random scale: $P^\omega$2 Under the scale condition

$P^\omega$3

fibrewise centering, and positivity of the asymptotic variance $P^\omega$4, the Birkhoff sums of $P^\omega$5 satisfy a quenched CLT under the random invariant measures $P^\omega$6. The variance is given by a Green–Kubo series, and $P^\omega$7 if and only if $P^\omega$8 is a coboundary (Dragičević et al., 2021).

A more abstract route is mixing of all orders for random dynamical systems. There the key innovation is a quenched stretched-exponential mixing estimate valid only when time gaps exceed a random threshold $P^\omega$9. Assuming $\sigma$00 for all $\sigma$01, the fibrewise Birkhoff sums

$\sigma$02

satisfy

$\sigma$03

for $\sigma$04-almost every $\sigma$05, with variance determined by an averaged Green–Kubo formula. In that framework, $\sigma$06 is equivalent to $\sigma$07 being a random $\sigma$08-coboundary (Auer, 29 May 2026).

Polymer and stochastic PDE models admit quenched Gaussian endpoint limits in weak disorder. For the mollified stochastic heat equation in $\sigma$09,

$\sigma$10

the renormalized Feynman–Kac formula defines a quenched polymer path measure. If $\sigma$11, then for almost every realization of the noise, the polymer endpoint is asymptotically diffusive: $\sigma$12 Under the rescaling $\sigma$13, this becomes a quenched CLT for the mollified stochastic heat equation (Broeker et al., 2017).

Several further examples show how broadly the concept applies. In a corner-growth setting, the energy of a typical up-right path in an independent environment satisfies a quenched CLT under a path-overlap condition $\sigma$14 with $\sigma$15; for uniformly chosen paths in a rectangle, $\sigma$16, so the paper requires $\sigma$17 on the environment moments. For the Ising model on random graphs, freezing the graph yields a random quenched CLT for magnetization with variance equal to susceptibility in the uniqueness regime, while averaging over graph randomness can create an additional Gaussian variance term. For the Ewens–Pitman model, a quenched functional CLT decomposes the fluctuations of the block count into sampling noise conditional on the asymptotic frequencies and fluctuations of the frequencies themselves, with a conditional independence structure given the $\sigma$18-diversity (Gromoll et al., 2018, Giardina' et al., 2014, Wang, 17 Mar 2026).

6. Structural themes, centering issues, and limitations

A persistent theme is that quenched CLT is governed by environment-specific bias as much as by variance growth. Random centering is sometimes indispensable. For Fourier transforms, the uncentered quenched CLT is equivalent to

$\sigma$19

for sums of stationary processes under Hannan’s condition, the centered quantity $\sigma$20 is the natural object; for one-dimensional RWRE, the position requires the environment-dependent correction $\sigma$21 even when the hitting-time CLT has deterministic normalization (Barrera et al., 2014, Volný et al., 2010, Ahn et al., 2017).

Normalization is equally model-sensitive. In classical diffusive settings the scale is $\sigma$22 or $\sigma$23, but the stationary linear-process counterexample shows that variance normalization by the true $\sigma$24 may support an annealed CLT while destroying quenched convergence, even under Maxwell–Woodroofe and Hannan conditions. Conversely, in planar random walk in random scenery the recurrent geometry forces the nonstandard normalization $\sigma$25, and the limit remains quenched Brownian motion with variance $\sigma$26 (Volny et al., 2015, Guillotin-Plantard et al., 2013).

The quenched CLT often sits inside a hierarchy of stronger statements. Functional forms yield Brownian limits in path space for Galton–Watson trees, random sceneries, random fields, expanding cocycles, and Ewens–Pitman partitions. Local forms identify pointwise heat-kernel asymptotics in balanced time-dependent environments. Rate results in one-dimensional RWRE show that even after the qualitative theorem is known, the environment may impose polynomial or logarithmic limits on Berry–Esseen decay (Bowditch, 2017, Deuschel et al., 2017, Ahn et al., 2020).

Failure mechanisms are now well documented. Annealed CLT may hold without quenched CLT; weak invariance principle may fail even when a variance-normalized annealed CLT holds; and coboundary structure alone does not control conditional bias. On the positive side, many successful proofs share a common triad: stabilization of conditional quadratic variation or correlations, ergodicity of the environment seen from the process, and suppression of the environment-specific drift by martingale centering, correctors, or conditional-mean estimates. This suggests that the quenched central limit theorem is less a single theorem than a family of fixed-environment Gaussian limit principles unified by conditioning but differentiated by how each model controls its residual dependence on the frozen medium (Peligrad, 2022, Volny et al., 2015, Auer, 29 May 2026).