pair density for various $w$’s at $\Delta=0$. From the scaling ansatz , the pair density at $\Delta = 0$ should collapse as $$t^{\delta} \rho_p(t) = {\cal G}(w^{\mu_\|} t ).
\label{Eq:mucoll}$$ With $\mu_\| \simeq 0.97$, all curves for the pair density are collapsed into a single curve as Fig. \[Fig:mucollapse\] shows. Next, we take $\Delta = \Delta_c(w)$ along the critical line. Since $\Delta_c(w) \simeq w^{1/\phi}$ and $\nu_\|/\phi = \mu_\|$, the scaling function should take the form $$t^{\delta} \rho_p(\Delta_c(w);t) = {\cal H}(w^{\mu_\|} t ),
\label{Eq:DCcoll}$$ where ${\cal H}(x)$ approaches a constant as $x \rightarrow
\infty$. In Fig. \[Fig:pair\], all curves at different critical points collapse well into a single curve. Hence we conclude that the scaling function Eq.  correctly describes the crossover behavior from the DP${}_\textrm{IM}$ to the the DP.

![\[Fig:pair\] (Color online) Scaling plot of $\rho_p(t) t^{\delta}$ vs $w^{\mu_\|}t$ with $\delta = 0.1595$ and $\mu_\|=0.97$ at criticality in semi-log scales. Inset: Scaling function ${\cal H}$ at finite $w$ and $w=0$. The asymptotic value of $w=0$ is different from the $w\rightarrow 0$ limit. ](ParkFig5.eps){width="45.00000%"}

![\[Fig:single\] (Color online) Log-log plot of the particle density for some finite $w$’s at $\Delta = 0$. For any finite value of $w$, the particle density $\rho(t)$ will 