monopoles spontaneously recombine to eliminate themselves from the sample. The plots indicate that the half life for monopole recombination in [Ho$_{2}$Ti$_{2}$O$_{7}$]{} would be approximately 150 seconds (much shorter than the equilibrium relaxation time) and suggests that equilibrium in the monopole density is reached at long times. Using the above estimate for the initial monopole density $n(t = 0) \sim 10^{-3}$, we recover a nominal equilibrium density of $n_{\rm eq} = 10^{-5}$ (per rare earth atom). Although this estimate is an upper limit it is nevertheless far from the expected equilibrium density, $n_{\rm eq}^{\rm 200~mK} \sim 10^{-13}$ (calculated by the method of Ref. , see Supplementary Fig. 1). It continues to evolve with temperature, being lower by a further order of magnitude at $T = 80$ mK (Fig. 3f). Most likely, the actual equilibrium monopole density is amplified by defects and disorder in the sample.

[**Magnetothermal avalanches**]{}

![image](figure4.pdf){width="12cm"}

Fig. 4 illustrates the effect of $t_{\rm w}$ on the magnetothermal avalanches. These occur when the injected power ($\mu_0 H \cdot J_{\rm m}$) overwhelms the extraction of thermal energy from the sample to the heat bath [@Jackson] such that monopoles are excited in great excess as the temperature steeply rises. The faster and 