fields for increasing $t_{\rm w}$ due to the smaller initial monopole density at the start of the field ramp. Also shown in the figure are the curves that result from slow conventional zero field cooling (CC) from 900 mK to 80 mK (at 1 mK.s$^{-1}$) followed by a 1000 s wait period. For the $^{162}$Dy and $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples the CC avalanche field is offset to higher fields, well outside the distribution of $H_{\rm ava}(t_{\rm w})$. For the $^{163}$Dy sample the CC curves falls within the distribution but near the long wait time curves. Also, we note for $^{163}$Dy, which happens to have better thermal contact, and thus faster cooling during the AQP, the CC curve again falls outside the distribution (shown in Supplementary Fig. 10 b). Thus slow cooling is more efficient at approaching equilibrium in [Dy$_{2}$Ti$_{2}$O$_{7}$]{} than is the AQP cooling followed by a long $t_{\rm w}$, especially for the low nuclear moment samples. This is typical behaviour for frustrated or disordered systems because slow cooling allows the system time to explore all available phase space.

Fig. 4c shows a much greater effect of $t_{\rm w}$ for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} with a larger spread of fields, saturating near 0.32 T 