Random-walk scaling without Bernoulli statistics: transport, persistence and resonances in the triangle map
Abstract
The triangle map is an area-preserving, piecewise-parabolic map of the torus with vanishing Lyapunov exponent. We study the symbolic process Xn=±1 given by the two halves of the torus, whose partial sums Sn are the transport of the lifted momentum, with an integrator free of accumulated rounding errors and exact computations on polygonal cylinder sets. We prove that the Koopman operator has no non-constant H1 eigenfunctions, that the symbol is not an H1 coboundary, and that runs of equal symbols are uniformly bounded. For generic parameters the walk is normally diffusive and Gaussian, but its diffusion coefficient is not that of a coin: away from the edges of the parameter square it varies between $0.2$ and $14$ times the Bernoulli value, and for a benchmark pair, with D=3.09±0.01, the excess is produced by the order of the bounded inter-event intervals, not by their distribution. The first-passage exponent is 1/2 with a non-Bernoulli amplitude, and worst-case Birkhoff averages decay as for a uniquely ergodic system. For β=0 the Green-Kubo sum cancels, D=0, yet Var(Sn) keeps growing slowly, sign changes are rare and the correlations of Fourier modes decay as n−0.2. We identify the zero-sum words, which become pure twists at β=0, as resonances that trap orbits with a computable 1/ℓ law, set the crossover time β−1/2 at small β, and, in an asymmetric deformation of the symbol, produce a logarithmic term in the variance whose coefficient we compute exactly. We formulate conjectures on unique ergodicity, mixing, the central limit theorem and the cohomology of the symbol.
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- Some Expressions involving Bernoulli, Harmonic, and Stirling Numbers
- TV between Bernoulli products, up to constants
BibTeX
@misc{esposti2026randomwalk,
title = {{Random-walk scaling without Bernoulli statistics: transport, persistence and resonances in the triangle map}},
author = {Mirko Degli Esposti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34179},
eprint = {2610.34179},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34179},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Random-walk scaling without Bernoulli statistics: transport, persistence and resonances in the triangle map
AU - Mirko Degli Esposti
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34179
UR - https://arxiv.org/abs/2610.34179
AB - The triangle map is an area-preserving, piecewise-parabolic map of the torus with vanishing Lyapunov exponent. We study the symbolic process X_n=±1 given by the two halves of the torus, whose partial sums S_n are the transport of the lifted momentum, with an integrator free of accumulated rounding errors and exact computations on polygonal cylinder sets. We prove that the Koopman operator has no non-constant H^1 eigenfunctions, that the symbol is not an H^1 coboundary, and that runs of equal symbols are uniformly bounded. For generic parameters the walk is normally diffusive and Gaussian, but its diffusion coefficient is not that of a coin: away from the edges of the parameter square it varies between $0.2$ and $14$ times the Bernoulli value, and for a benchmark pair, with D=3.09±0.01, the excess is produced by the order of the bounded inter-event intervals, not by their distribution. The first-passage exponent is 1/2 with a non-Bernoulli amplitude, and worst-case Birkhoff averages decay as for a uniquely ergodic system. For β=0 the Green-Kubo sum cancels, D=0, yet Var(S_n) keeps growing slowly, sign changes are rare and the correlations of Fourier modes decay as n^−0.2. We identify the zero-sum words, which become pure twists at β=0, as resonances that trap orbits with a computable 1/ℓ law, set the crossover time β^−1/2 at small β, and, in an asymmetric deformation of the symbol, produce a logarithmic term in the variance whose coefficient we compute exactly. We formulate conjectures on unique ergodicity, mixing, the central limit theorem and the cohomology of the symbol.
ER -