Figure 1

Magnetization curves $M_R(H)$ for disorders $R=0.52-0.76$ and system size $L=131072$ for which $R_c^{\mathrm{eff}}=0.605$—see Eq. (10). For $R<R_c^{\mathrm{eff}}$, the value $H_{\mathrm{sp}}$ of external magnetic field at which spanning avalanche occurs varies with random field configuration (RFC). Main panel: magnetization curves for single RFC sorted in increasing disorder $R$. Inset: magnetization curves averaged over 30 RFC; while this number is small, steps (which appear due to spanning avalanches and stochastic distribution of $H_{\mathrm{sp}}$) are visible.Reuse & Permissions

Figure 2

Scaling collapse of magnetization $M$ (main panel) and susceptibility $\chi$ curves (inset) for disorders $R=0.70–0.76$ and system size $L=131072$. The curves are averages of 30 random field configurations for each $R$. Collapsing parameters: $\beta =0.15$, $\beta \delta =4.8$, $R_c=0.54$, $H_c=1.275$, $M_c=0$, and $b=0.24$; for both collapses, the width (see 11) is $w=2.1\times {10}^{-2}$.Reuse & Permissions

Figure 3

Scaling collapse of integrated distributions $D_R^{(\mathrm{int})}(S)$ of avalanche size $S$ (symbols) for disorders $R=0.64–0.70$ and system size $L=65536$; $w=4.5\times {10}^{-2}$. The curves are averages of 600 random field configurations for each $R$. The solid line represents the phenomenological curve (5). Collapsing of binned distributions $D_R^{(\mathrm{bin})}(S)$ for the same simulation data is shown in the top inset; $w=9.4\times {10}^{-3}$. In the bottom inset, integrated distributions $D_R^{(\mathrm{int})}(S)$ of avalanche size $S$ for system sizes $L=8192-131072$ are shown. The graphs for $L<131072$ are shifted up for better visibility.Reuse & Permissions

Figure 4

Power-law divergence $\xi \sim |r{|}^{-\nu }$ of correlation length $\xi$ with reduced disorder $r$ for reduced magnetic field $h^{\prime }=0$. In the bottom inset, the same data are shown against $R$ on a linear scale. Top inset: scaling collapse of correlation function $G_R^{(\mathrm{int})}(x)$ for disorders $R=0.64–0.90$ and system size $L=131072$; $w=6.8\times {10}^6$. The curves are averages of 30 random field configurations for each $R$.Reuse & Permissions

Figure 5

Scaling collapse of the correlation function $G_{R,H}(x)$ for the avalanches at reduced field $h^{\prime }=0$. The collapse is obtained for $\eta =1$ and correlation lengths from Fig. 4; $w=4.6\times {10}^6$. Inset: the same collapse on the lin-log scale illustrates the applicability of approximation (8).Reuse & Permissions

Figure 6

Effective critical disorder $R_c^{\mathrm{eff}}(L)$ versus system size $L$ (symbols) and the power-law prediction $[R_c^{\mathrm{eff}}(L)-R_c]/R_c^{\mathrm{eff}}(L)\sim L^{-1/\nu }$ (continuous line) with $R_c=0.54$ and $\nu =5.14$. Top inset: Bray-Moore scaling $\xi \sim e^{\tilde{a}/|R_c^{\mathrm{eff}}(L)-R_c{|}^2}$, $R_c\sim 0$; $\Delta R_c^{\mathrm{eff}}(L)=0.00125$. Bottom inset: size distributions for $R=0.38–0.55$, $L=65536$, and more than 600 random field configurations for each $R$.Reuse & Permissions