guessed. We do this by choosing the pairing with the minimal sum of connection path lengths using `Blossom V` [@Kolmogorov2009], which is the latest improvement on Edmonds’ minimal-weight perfect matching algorithm [@Edmonds1965]. If many anyons are present, using the complete graph as the input to this algorithm is numerically infeasible. In excellent approximation, we therefore replace the complete graph by a Delaunay triangulation [@footnote03].

As a useful reference, we show in Fig. \[nonint\_fig\] numerical results for the non-interacting system, i.e., $A = 0$. The relevant rates entering Eq. are $\gamma(0)$ (rate for an anyon to hop to a free neighboring site), $\gamma(-2J)$ (rate to create an anyon pair) and $\gamma(2J) = \gamma(-2J)e^{2J\beta}$ (rate to annihilate a pair of adjacent anyons, obtained from the detailed balance condition). Figure \[nonint\_fig\] illustrates the typical behavior of $\langle Z \rangle$ and $\langle Z_{\rm ec} \rangle$, in agreement with previous literature [@Alicki2009; @Nussinov2008; @Hamma2009; @Bravyi2009; @Kay2008]. We refer to Sec. \[sec:lifetime\] for a more detailed discussion.

Diffusion of anyons and memory lifetime {#sec:lifetime}
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It is the purpose of this section to establish a formula for the lifetime of the quantum memory. A static criterion was discussed in Ref. [@Dennis2002]: assuming independent errors, the toric 