KPZ fluctuation exponent for ballistic deposition: $\mathrm{Var}\,\xi_t(0)\asymp t^{2/3}$

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KPZBallistic.kpz_variance_exponent

by Lucas · Oct 11, 2026 · Mathlib 0df444a (Lean v4.33.1)

ballistic-depositionkpzprobability

Let $\xi_t(0)$ be the height of column $0$ at time $t$ in one-dimensional (nearest-neighbour) ballistic deposition on $\mathbb Z$ , started from a flat surface and driven by independent rate-one Poisson clocks. There are constants $0<c\le C<\infty$ and $T$ such that

c\,t^{2/3}\le \operatorname{Var}\xi_t(0)\le C\,t^{2/3}\qquad\text{for all } t\ge T.

This is the KPZ prediction that the column height fluctuates around its mean on the scale $t^{1/3}$ ; it is open. The best known bounds are $\operatorname{Var}\xi_t(0)\gtrsim\log t$ (Penrose 2008) and $\operatorname{Var}\xi_t(0)\lesssim t$ (Sudijono 2026).

Formalization Note Mathlib's variance is $0$ for variables that are not square-integrable, so the lower bound also forces square-integrability.

Preamble

import Mathlib
import Definitions.Def_KPZBallistic_Defs

open MeasureTheory ProbabilityTheory Filter Topology

Formal statement

namespace KPZBallistic

theorem kpz_variance_exponent {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
    [IsProbabilityMeasure P] (τ : ℤ → ℕ → Ω → ℝ) (hτ : IsPoissonClocks P τ) :
    ∃ c C T : ℝ, 0 < c ∧ 0 < C ∧ ∀ t ≥ T,
      c * t ^ ((2 : ℝ) / 3) ≤ variance (bdHeightReal τ t 0) P ∧
      variance (bdHeightReal τ t 0) P ≤ C * t ^ ((2 : ℝ) / 3) := by sorry

end KPZBallistic

Source

Open Problem Garden, 'KPZ Universality Conjecture' (posted by T. Kojar, 2020), http://www.openproblemgarden.org/op/kpz_universality_conjecture; I. Corwin, The Kardar-Parisi-Zhang equation and universality class, Random Matrices Theory Appl. 1 (2012), https://arxiv.org/abs/1106.1596, Section 1.1.1, p. 3 ('in time t the column height will fluctuate around its mean like t^{1/3} ... This prediction has not been proved'); T. Sudijono, A Linear Variance Bound for Ballistic Deposition (2026), https://arxiv.org/abs/2609.24056, Section 1, p. 2 ('Var f(t,0) is expected to scale as t^{2/3} when d = 1')

Human review

Endorsed by Shuze Chen · Oct 11, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 11, 2026
Confirmed by the mission captain (proposal self-audit).

View graph

KPZ fluctuation exponent for ballistic deposition: Var ξt(0)≍t2/3\mathrm{Var}\,\xi_t(0)\asymp t^{2/3}Varξt​(0)≍t2/3

KPZ fluctuation exponent for ballistic deposition: $\mathrm{Var}\,\xi_t(0)\asymp t^{2/3}$