of Mott-like correlations [@sherson2010single; @larson2008mott; @katsimiga2017dark] in the system \[see also the corresponding discussion in Sec. \[def\_cor\_functions\]\] and suggests the tendency of the observed filaments to be localized structures. Moreover, we can infer that each $\sigma$-species fermion, and especially the $A$-species ones, is more likely to be localized in one filament and do not reside in two or more filaments.

We next study the two-body intraspecies correlation function $g_{AA}^{(2)}(x,x';t)$, see Figs. \[abb:coh3-1\] (c~1~)-(c~4~). A strongly anti-correlated behavior within each filament, see the depleted diagonal behavior, occurs for every time instant \[e.g. $g_{AA}^{(2)}(x=-2.5,x^\prime=-2.5;t=24)\approx0$ in Fig. \[abb:coh3-1\] (c~1~)\], while two different filaments appear to be correlated \[e.g. $g_{AA}^{(2)}(x=-2.5,x^\prime=2.5;t=24)\approx1.3$ in Fig. \[abb:coh3-1\] (c~1~)\]. As a consequence, two particles of the $A$-species cannot reside in the same filament but they are more likely to be found in any pair of distinct filaments. The corresponding interspecies correlation function $g_{AB}^{(2)}(x,x')$, shown in Figs. \[abb:coh3-1\] (d~1~)-(d~4~), displays similar characteristics to $g_{AA}^{(2)}(x,x';t)$. Namely, a correlation hole exists \[see e.g. $g_{AB}^{(2)}(x=-2.5,x^\prime=-2.5;t=24)\approx0$ in Fig. \[abb:coh3-1\] (d~1~)\] which excludes the possibility of an $A$ and a $B$ particle to be in the same filament. However, the off-diagonal elements of $g_{AB}^{(2)}(x,x';t)$ exhibit a correlated behavior \[see e.g. $g_{AB}^{(2)}(x=-2.5,x^\prime=2.5;t=24)\approx2.2$ in Fig. \[abb:coh3-1\] (d~1~)\] 