  We study the XXZ model with a random magnetic field in contact with a weakly
disordered spin chain, acting as a finite thermal bath. We revise Fermi's
golden rule description of the interaction between the thermal bath and the XXZ
spin chain, contrasting it with a nonperturbative quantum avalanche scenario
for the thermalization of the system. We employ two-point correlation functions
to define the extent $\xi_d$ of the thermalized region next to the bath.
Unbounded growth of $\xi_d$ proportional to the logarithm of time or faster is
a signature of an avalanche. Such behavior signifies the thermalization of the
system, as we confirm numerically for a generic initial state in the ergodic
and critical regimes of the XXZ spin chain. In the many-body localized regime,
a clear termination of avalanches is observed for specifically prepared initial
states and, surprisingly, is not visible for generic initial product states.
Additionally, we extract the localization length of the local integrals of
motion and show that a bath made out of a weakly disordered XXZ chain has a
similar effect on the system as a bath modeled by a Hamiltonian from a Gaussian
orthogonal ensemble of random matrices. We also comment on the result of the
earlier study (Phys. Rev. B 108, L020201 (2023)), arguing that the observed
thermalization is due to external driving of the system and does not occur in
the autonomous model. Our work reveals experimentally accessible signatures of
quantum avalanches and identifies conditions under which termination of the
avalanches may be observed.
