Eq. (\[T\_estimate\]) (dotted curve) gives good agreement for $f_c\simeq 0.1$.](nonint_lifetime_vs_T.pdf){width="0.95\columnwidth"}

Direct and indirect diffusion of anyons {#app:indirect}
---------------------------------------

To estimate the error creation rate, we first study the diffusive motion of anyons in the non-interacting model. To determine the diffusion constant $D$, we consider an isolated anyon in the lattice and its probability $p_{i,j}$ to be at site $(i,j)$. In the Ohmic case, we have $\gamma(0)\neq 0$, and direct hopping to neighboring sites is thus allowed. In the continuum limit, a standard diffusion equation $\frac{dp({\bf r})}{dt}=D\nabla^2 p({\bf r})$ with $D=\gamma(0)$ is obtained. The resulting decay of the bare and error-corrected logical operators in the simple case of a single pair is discussed in Appendix \[app:single\_pair\].

For a super-Ohmic bath where $\gamma(0)=0$, diffusion is still possible due to ‘indirect hopping’. We assume $2\beta J \gg 1$, such that, since $\gamma(2J)=e^{2\beta J}\gamma(-2J)$, the recombination of a pair of anyons is essentially instantaneous. Hopping from the site $(i, j)$ to, e.g., $(i,j+2)$ is possible by creation of an anyon pair occupying sites $(i,j+1)$ and $(i,j+2)$. This event occurs with rate $\gamma(-2J)$. Since the intermediate state can decay back to the initial state, the actual rate for the indirect hopping process is $\gamma(-2J)/2$. Similar 