The Z-Dirac and massive Laplacian operators in the Z-invariant Ising model

摘要

Consider an elliptic parameter $k$. We introduce a family of $Z^u$-Diracoperators $(mathsf{K}(u))_{uinRe(mathbb{T}(k))}$, relate them to the$Z$-massive Laplacian of [BdTR17b], and extend to the full $Z$-invariant casethe results of Kenyon [Ken02] on discrete holomorphic and harmonic functions,which correspond to the case $k=0$. We prove, in a direct statistical mechanicsway, how and why the $Z^u$-Dirac and $Z$-massive Laplacian operators appear inthe $Z$-invariant Ising model, considering the case of infinite and finiteisoradial graphs. More precisely, consider the dimer model on the Fisher graph${mathsf{G}}^{scriptscriptstyle{mathrm{F}}}$ corresponding to a$Z$-invariant Ising model. Then, we express coefficients of the inverse FisherKasteleyn operator as a function of the inverse $Z^u$-Dirac operator and alsoas a function of the $Z$-massive Green function. This proves a (massive) randomwalk representation of important observables of the Ising model. We prove thatthe squared partition function of the Ising model with + boundary conditions isequal, up to a constant, to the determinant of the $Z$-massive Laplacianoperator with specific boundary conditions, the latter being the partitionfunction of rooted spanning forests. In proving these results, we relate theinverse Fisher Kasteleyn operator and that of the dimer model on the bipartitegraph ${mathsf{G}}^{scriptscriptstyle{mathrm{Q}}}$ arising from theXOR-Ising model, and we prove matrix relations between the Kasteleyn matrix of${mathsf{G}}^{scriptscriptstyle{mathrm{Q}}}$ and the $Z^u$-Dirac operator,that allow to reach inverse matrices as well as determinants.