Growing-time trace fluctuations and self-averaging form factors of Grover walks on random lifts
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, $d\geq3$. 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 $t\geq1$, it suffices that $M\to\infty$ and $R^4(d-1)^R/N\to0$, 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.