Growing-time trace fluctuations and self-averaging form factors of Grover walks on random lifts
Abstract
We prove a functional central limit theorem for trace statistics of Grover walks on random N-lifts of a fixed finite connected simple d-regular graph, d≥3. All observations are computed from the same random lift. The squared centred traces, divided by their observation times, have averages converging in probability to a degree-dependent constant. Their partial sums, centred at this constant and divided by the square root of the window length, converge to Brownian motion. For a window of length M beginning at t≥1, it suffices that M→∞ and R4(d−1)R/N→0, where R=t+M. The initial time may remain fixed, with no further restriction on M/t. Convergence holds jointly with fixed low-degree polynomial trace statistics; their limiting vector is independent of the Brownian motion. For non-bipartite bases, we also identify Gaussian trace limits at fixed time offsets. Under common initial-time normalisation, subtracting a deterministic growth profile gives an explicit Brownian time change when M/t has a finite limit. Bipartite bases retain the same averaged constant and have twice the Brownian variance rate. The proof combines a growing-dimensional Poisson approximation, a uniform bilinear Chebyshev-energy estimate, a time-decaying Poisson--Gaussian coupling, and a Gaussian quadratic-array theorem. Separate moment estimates give expectation-centred versions under an additional sufficient condition.
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BibTeX
@misc{nagata2026growingtime,
title = {{Growing-time trace fluctuations and self-averaging form factors of Grover walks on random lifts}},
author = {Nino Nagata and Ryoichi Suzuki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36320},
eprint = {2610.36320},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36320},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Growing-time trace fluctuations and self-averaging form factors of Grover walks on random lifts
AU - Nino Nagata
AU - Ryoichi Suzuki
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.36320
UR - https://arxiv.org/abs/2610.36320
AB - We prove a functional central limit theorem for trace statistics of Grover walks on random N-lifts of a fixed finite connected simple d-regular graph, d≥3. All observations are computed from the same random lift. The squared centred traces, divided by their observation times, have averages converging in probability to a degree-dependent constant. Their partial sums, centred at this constant and divided by the square root of the window length, converge to Brownian motion. For a window of length M beginning at t≥1, it suffices that M→∞ and R^4(d−1)^R/N→0, where R=t+M. The initial time may remain fixed, with no further restriction on M/t. Convergence holds jointly with fixed low-degree polynomial trace statistics; their limiting vector is independent of the Brownian motion. For non-bipartite bases, we also identify Gaussian trace limits at fixed time offsets. Under common initial-time normalisation, subtracting a deterministic growth profile gives an explicit Brownian time change when M/t has a finite limit. Bipartite bases retain the same averaged constant and have twice the Brownian variance rate. The proof combines a growing-dimensional Poisson approximation, a uniform bilinear Chebyshev-energy estimate, a time-decaying Poisson--Gaussian coupling, and a Gaussian quadratic-array theorem. Separate moment estimates give expectation-centred versions under an additional sufficient condition.
ER -