Search

Directional zero–one laws beyond iid environments and iid ballisticity

A walk in a random environment escapes in a given direction with probability 0 or 1, and escape forces positive speed.

3 papers September 23, 2026 Explained by AI

Picture a walker on a grid whose step odds are set at random, once, at every site. One question asks whether the chance of drifting off in a fixed direction must be 0 or 1. Another asks whether drifting off forces a positive average speed. These papers answer yes in key cases. Lean has checked the iid zero–one law and the speed result; the dependent-environment extension is not checked.

First step
1981
Kalikow, the first directional transience results
Significance
Major
It settles two central problems for random walks in iid random environments, with the main statements checked in Lean.
Trust
Formal proof listed
Field
Probability and statistical mechanics
3 papers

The explanation

Start on a line of sites. At each site, flip a fair coin once and keep the result forever. Heads makes that site push right with chance 0.9. Tails makes it push right with chance only 0.3. A walker then steps left or right using the odds at its current site. Does it drift off to the right? A classical test by Solomon looks at the ratio of left to right odds. The ratios here are 1/9 and 7/3. The average of their logarithms is about −0.675, which is negative. So the walker escapes to the right. But the plain average of the ratios is about 1.22, at least 1. That makes its long-run speed zero. Traps of left-pushing sites slow it so much that it covers distance more slowly than any fixed rate.

This is a random walk in a random environment. The environment is the list of step odds at every site, drawn once and then held fixed. Directional transience means the walker eventually leaves every half-space behind it in a chosen direction. Ballisticity asks for more: steady progress at a positive speed. The line example shows the two can differ, already in one dimension.

Two old questions sit here for higher dimensions. First, can the chance of escaping in one direction lie strictly between 0 and 1? Kalikow, with Sznitman and Zerner, showed only that escape in one of the two opposite directions has chance 0 or 1. In two dimensions, Zerner and Merkl proved the full zero–one law. Their argument uses crossings of paths in the plane. Second, does escape in one direction force positive speed? Fribergh and Kious stated this as a conjecture in 2016.

The second paper takes the first question. Averaging over the environment makes the walk remember. Returning to a site reveals more about its odds. The paper proves the zero–one law in every dimension from 3 up. The environment must be iid, meaning each site's odds are drawn independently from one fixed recipe. Every step must have positive chance, but no common lower bound is needed.

The third paper takes the second question. Here the odds must be uniformly elliptic: every step has chance at least some fixed κ above 0. It proves that escape in one direction gives a fixed limiting velocity, pointing into that direction. This holds in every dimension from 2 up. The obstacle was that walks split into independent pieces only gives finite pieces. It does not show their average length is finite. The proof first controls how far a piece wanders and only later how long it lasts.

The first paper extends the zero–one law beyond independence. Sites may depend on each other, but only within a fixed distance. Correlations that merely fade are not enough. Bramson, Zeitouni and Zerner built environments with slowly fading correlations where escape in each of two opposite directions has positive chance.

How sure is this? A computer proof checker called Lean has checked formal versions of the iid zero–one law under strict ellipticity, for dimension 3 and up. It has also checked that escape implies a positive velocity in dimension 2 and up, under uniform ellipticity. The finite-range dependent extension is not formalized. No specialist review has been published.

What it doesn't show. This does not give the zero–one law in dimension 2 beyond earlier work, or for environments with long-range dependence.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major. The iid zero–one law and the speed result have Lean checks; the finite-range extension does not, and no specialist review is published.

Words used

Random environment
Step odds placed at every site of a grid by a random draw, then held fixed while the walker moves.
Directional transience
The walker eventually moves off forever in a chosen direction, leaving every half-space behind it.
Ballisticity
Moving off in a direction at a positive average speed, not just eventually.
Zero–one law
A result saying an event has probability exactly 0 or exactly 1, never anything in between.
Ellipticity
Every allowed step has positive chance; uniform ellipticity adds a common lower bound for those chances.
iid
Independent and identically distributed: each site's odds are drawn separately from the same recipe.
Finite-range dependence
Sites far enough apart have independent odds; nearby sites may be linked.

The original papers (3)

Lean notes

All quoted lines (12)
  1. Directional transience says that a random walk eventually leaves every backward half-space.
    Directional transience implies ballisticity Back to text
  2. The distinction between escape and speed is essential: already in one dimension, uniformly elliptic iid environments can give a transient walk with zero speed; see Solomon and the discussion in.
    Directional transience implies ballisticity Back to text
  3. Kalikow's work, together with the regeneration formulation of Sznitman and Zerner, established the sign-union law for uniformly elliptic iid environments, P0(Aℓ∪A−ℓ)∈{0,1}.P_0(A_\ell\cup A_{-\ell})\in\{0,1\}.
    A directional zero–one law for finite-range-dependent random environments Back to text
  4. The planar arguments exploit intersections of paths.
    A directional zero–one law for finite-range-dependent random environments Back to text
  5. Fribergh and Kious formulate it with precisely the one-direction transience assumption used here.
    Directional transience implies ballisticity Back to text
  6. Averaging over an independent environment at every lattice site creates a different process: revisiting a site also revisits information about its transition probabilities.
    A directional zero–one law under strict ellipticity Back to text
  7. We resolve this problem positively in every dimension d≥3d\ge3 for iid nearest-neighbor environments with strictly positive transition probabilities.
    A directional zero–one law under strict ellipticity Back to text
  8. No common positive lower bound on those probabilities is assumed.
    A directional zero–one law under strict ellipticity Back to text
  9. We prove that they coincide for nearest-neighbor walks in iid uniformly elliptic environments in every dimension at least two.
    Directional transience implies ballisticity Back to text
  10. The proof here first controls a piece's spatial radius—its greatest distance from its starting point— and only later its duration.
    Directional transience implies ballisticity Back to text
  11. Here we extend the uniformly elliptic case to finite-range dependence.
    A directional zero–one law for finite-range-dependent random environments Back to text
  12. Exact finite-range independence is also stronger than decay of local correlations: Bramson, Zeitouni, and Zerner constructed stationary, uniformly elliptic, polynomially mixing environments in dimension at least three with positive probability of escape in each of two opposite directions.
    A directional zero–one law for finite-range-dependent random environments Back to text

Other sources

  1. openai/math, Lean scope note for family 220 For an i.i.d. uniformly elliptic nearest-neighbor random environment on Zd\mathbb Z^d, the formalization proves that almost-sure directional transience implies convergence of Xn/nX_n/n to a deterministic velocity with positive projection in that direction for every d≥2d\ge2.
  2. openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model
  3. F. Solomon, “Random walks in a random environment”, The Annals of Probability 3 (1975), 1–31
  4. A. Fribergh and D. Kious, “Local trapping for elliptic random walks in random environments in ℤ^d”, Probability Theory and Related Fields 165 (2016), 795–834

Browse Probability and statistical mechanics Open since the 1980sLean-checked results