  The dependence of the chaotic phase of the Bose-Hubbard Hamiltonian on
particle number $N$, system size $L$ and particle density is investigated in
terms of spectral and eigenstate features. We analyze the development of the
chaotic phase as the limit of infinite Hilbert space dimension is approached
along different directions, and show that the fastest route to chaos is the
path at fixed density $n \lesssim 1$. The limit $N \to \infty$ at constant $L$
leads to a slower convergence of the chaotic phase towards the random matrix
theory benchmarks. In this case, from the distribution of the eigenstate
generalized fractal dimensions, the ergodic phase becomes more distinguishable
from random matrix theory for larger $N$, in a similar way as along
trajectories at fixed density.
