  Several works have observed heavy-tailed behavior in the distributions of
returns in different markets, which are observable indicators of underlying
complex dynamics. Such prior works study return distributions that are
marginalized across the individual stocks in the market, and do not track
statistics about the joint distributions of returns conditioned on different
stocks, which would be useful for optimizing inter-stock asset allocation
strategies. As a step towards this goal, we study emergent phenomena in the
distributions of returns as captured by their pairwise correlations. In
particular, we consider the pairwise (between stocks $i,j$) partial
correlations of returns with respect to the market mode, $c_{i,j}(\tau)$,
(thus, correcting for the baseline return behavior of the market), over
different time horizons ($\tau$), and discover two novel emergent phenomena:
(i) the standardized distributions of the $c_{i,j}(\tau)$'s are observed to be
invariant of $\tau$ ranging from from $1000 \textrm{min}$ (2.5 days) to $30000
\textrm{min}$ (2.5 months); (ii) the scaling of the standard deviation of
$c_{i,j}(\tau)$'s with $\tau$ admits \iffalse within this regime is empirically
observed to \fi good fits to simple model classes such as a power-law
$\tau^{-\lambda}$ or stretched exponential function $e^{-\tau^\beta}$
($\lambda,\beta > 0$). Moreover, the parameters governing these fits provide a
summary view of market health: for instance, in years marked by unprecedented
financial crises -- for example $2008$ and $2020$ -- values of $\lambda$
(scaling exponent) are substantially lower. Finally, we demonstrate that the
observed emergent behavior cannot be adequately supported by existing
generative frameworks such as single- and multi-factor models. We introduce a
promising agent-based Vicsek model that closes this gap.
