  We present results on a meso-scale model for amorphous matter in athermal,
quasi-static (a-AQS), steady state shear flow. In particular, we perform a
careful analysis of the scaling with the lateral system size, $L$, of: i)
statistics of individual relaxation events in terms of stress relaxation, $S$,
and individual event mean-squared displacement, $M$, and the subsequent load
increments, $\Delta \gamma$, required to initiate the next event; ii) static
properties of the system encoded by $x=\sigma_y-\sigma$, the distance of local
stress values from threshold; and iii) long-time correlations and the emergence
of diffusive behavior. For the event statistics, we find that the distribution
of $S$ is similar to, but distinct from, the distribution of $M$. We find a
strong correlation between $S$ and $M$ for any particular event, with $S\sim
M^{q}$ with $q\approx 0.65$. $q$ completely determines the scaling exponents
for $P(M)$ given those for $P(S)$. For the distribution of local thresholds, we
find $P(x)$ is analytic at $x=0$, and has a value $\left.
P(x)\right|_{x=0}=p_0$ which scales with lateral system length as $p_0\sim
L^{-0.6}$. Extreme value statistics arguments lead to a scaling relation
between the exponents governing $P(x)$ and those governing $P(S)$. Finally, we
study the long-time correlations via single-particle tracer statistics. The
value of the diffusion coefficient is completely determined by $\langle \Delta
\gamma \rangle$ and the scaling properties of $P(M)$ (in particular from
$\langle M \rangle$) rather than directly from $P(S)$ as one might have naively
guessed. Our results: i) further define the a-AQS universality class, ii)
clarify the relation between avalanches of stress relaxation and diffusive
behavior, iii) clarify the relation between local threshold distributions and
event statistics.
