Coarse endpoint concentration at every diverging scale for one-dimensional space-time random walks

Abstract

For a nearest-neighbor walk on Z in a fixed i.i.d. space-time environment with transition probabilities strictly between zero and one, we prove that its quenched and averaged endpoint laws agree, after coarse graining into deterministic intervals of any diverging length, outside an environment event of probability smaller than every inverse power of time. No uniform ellipticity or quantitative rate of divergence of the interval length is required. This is the affirmative one-dimensional assertion contrary to Conjecture 1 in Berger’s contribution to Oberwolfach Report 38/2009. The proof combines replica overlap estimates, environmental martingales, same-parity coalescence and concentration over a short final segment. Priority qualification: novelty and historical priority remain unresolved. A bounded literature audit established no exact full anticipation, but four relevant older original full texts could not be accessed. We make no claim of firstness, independent discovery or that the conjecture remained open until this work. Established late-layer and coalescing methods, including those of Dunlap–Gu and Barraquand–Rychnovsky, are explicitly credited.

The idea in plain English

Imagine a walker on the integers whose chance of stepping right is randomly chosen separately for every place and time. Once those chances are fixed, the walker still has its own randomness. The resulting endpoint distribution can be compared with the distribution obtained by averaging over all environments.

The paper groups possible endpoints into intervals and compares the probability assigned to every interval. If the interval length tends to infinity, however slowly, the total discrepancy becomes small. For any fixed positive tolerance, the chance of an environment violating that tolerance eventually falls below every prescribed inverse power of time.

The surprising feature is that no minimum growth rate for the intervals is required. The result also permits bias and transition probabilities arbitrarily close to zero or one, provided the environment law is fixed and neither endpoint occurs with positive probability.

This establishes the affirmative one-dimensional statement opposed by a conjecture in Berger’s 2009 workshop contribution. The note is explicit that its historical novelty remains unresolved; earlier local-limit and fluctuation work must receive credit, and several older full texts remain unexamined.

The precise result

Let the right-step probabilities pₛ,ₓ be iid over space and time, with 0 < pₛ,ₓ < 1 almost surely. A nearest-neighbor walk on Z starts at zero. Write μ_N^ω for its endpoint law in the fixed environment and α_N = Eμ_N^ω for its averaged endpoint law.

For every deterministic real sequence M_N > 0 tending to infinity, partition Z using the half-open intervals [b_N + jM_N, b_N + (j + 1)M_N), intersected with Z. For every ε, K > 0, the environmental probability that Σ_I |μ_N^ω(I) − α_N(I)| exceeds ε is at most C_{ε,K} N^(−K) for all sufficiently large N, uniformly in each deterministic alignment b_N.

The discrepancy is full ℓ¹ distance, twice the usual total variation distance. Constants may depend on the fixed environment law; the eventual threshold may also depend on the chosen scale sequence. The scale may diverge nonmonotonically and arbitrarily slowly.

The conclusion includes biased averaged walks, nearest-neighbor parity, and real interval lengths. It assumes neither uniform ellipticity nor inverse-moment bounds for the transition probabilities. All analytic estimates required by the proof are supplied in the note.

Why the result holds

Two walkers sharing the environment give a one-dimensional difference chain. Its return and occupation bounds control how much the endpoint law can depend on each newly exposed time layer.

Environmental martingales first give sharp concentration at an auxiliary polynomial interval scale. A same-parity coalescing coupling then erases differences within those auxiliary cells over a shorter final time segment.

The remaining environment is divided into independent moving strips. Before that segment, no relevant strip carries much starting mass; changing one strip therefore has small influence. Bounded differences concentrate the final coarse discrepancy around its conditional mean.

The target interval length enters only through an expectation estimate that tends to zero whenever it diverges. It need not grow at the auxiliary polynomial rate. A controlled path-killing error allows the independent-strip argument to return to the unrestricted walk.

Scope and limitations

The theorem concerns a fixed iid space-time environment law and nearest-neighbor jumps in one spatial dimension. It does not cover static or correlated environments, and its constants need not remain controlled as elliptic laws approach deterministic-arrow environments.

The tolerance ε and inverse-power exponent K are fixed before taking large time. The theorem does not give a uniform convergence rate for every diverging scale sequence or a bound for a shrinking tolerance.

The interval partition is deterministic. Uniformity in its deterministic alignment is not a simultaneous assertion for partitions selected after observing the environment.

This is a statement about coarse endpoint probabilities, not a pointwise Gaussian local limit theorem or a claim that microscopic environmental fluctuations disappear. A qualitative central or local limit theorem alone does not supply the stated tail bound.

Novelty and historical priority remain unresolved. The note could not inspect the complete original texts of Boldrighini–Minlos–Pellegrinotti (1999 and 2004), Boldrighini–Pellegrinotti (2001), or Bérard (2004); stronger estimates in those papers have not been excluded as antecedents. Firstness, independent discovery, a new method and present-day open status are not claimed.

Status and AI use

AI tools were used extensively in solving, drafting, and checking this work. The note has not undergone conventional human peer review. The author is responsible for the claims; finite checks supplement the analytic proof and do not certify novelty.

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Collaborate on this result

I welcome substantive collaboration on an independent probability-theoretic review of the all-scale concentration argument, or on resolving the remaining primary-literature comparisons before a journal submission. Expertise in space-time random environments, quenched local limits, and concentration inequalities would be particularly useful.

Authorship and responsibilities would be discussed in relation to the substantive contributions made.

Cite this paper

Alec Kriebel. (2026). Coarse endpoint concentration at every diverging scale for one-dimensional space-time random walks (Version 1.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23445342

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Background and prior work

• Berger, Quenched Local Limit Theorem for Space-time Random Walk in Random Environment (2009): Printed pages 2151–2152 give the higher-dimensional coarse concentration theorem and Conjecture 1, which predicts failure in one transverse dimension.
• Dunlap and Gu, A quenched local limit theorem for stochastic flows: A prior continuum local-limit theorem with a stationary random corrector. Its short final-layer method is explicitly credited; its model and conclusion differ from the discrete coarse ℓ¹ concentration statement here.
• Barraquand and Rychnovsky, Random walk on nonnegative integers in beta distributed random environment: Proposition 1.13 concerns a full-space beta-environment local limit. Its discussion and proof use prior final-layer and ordered-coalescence ideas; these mechanisms are not claimed as new here.
• Boldrighini, Minlos and Pellegrinotti, Random walks in quenched iid space-time random environment are always a.s. diffusive (2004): An important earlier quenched central-limit result. Its complete original proof was not available for comparison, so it has not been ruled out as containing stronger relevant estimates.
• Boldrighini and Pellegrinotti, T⁻¹/⁴-noise for random walks in dynamic environment on Z (2001): Earlier work on environment-dependent fluctuation corrections. It is one of the full-text gaps that prevent a certified novelty claim.
• Bérard, The almost sure central limit theorem for one-dimensional nearest-neighbour random walks in a space-time random environment (2004): Earlier nearest-neighbor quenched central-limit theory. The available abstract is not a substitute for checking all estimates in the original proof.