Return probability for the loop-erased random walk and mean height in sandpile : a proof

Single site height probabilities in the Abelian sandpile model, and the corresponding mean height $<h>$, are directly related to the probability $P_{\mathrm{r}\mathrm{e}\mathrm{t}}$ that a loop erased random walk passes through a nearest neighbour of the starting site (return probability). The exact values of these quantities on the square lattice have been conjectured, in particular $<h>=25\mathrm{/}8$ and $P_{\mathrm{r}\mathrm{e}\mathrm{t}}=5\mathrm{/}16$. We provide a rigourous proof of this conjecture by using a {\it local} monomer-dimer formulation of these questions.