Two-point boundary correlation functions of dense loop models

International audienceWe investigate six types of two-point boundary correlation functions in thedense loop model. These are defined as ratios $Z/Z^0$ of partition functions onthe $m\times n$ square lattice, with the boundary condition for $Z$ dependingon two points $x$ and $y$. We consider: the insertion of an isolated defect (a)and a pair of defects (b) in a Dirichlet boundary condition, the transition (c)between Dirichlet and Neumann boundary conditions, and the connectivity ofclusters (d), loops (e) and boundary segments (f) in a Neumann boundarycondition.For the model of critical dense polymers, corresponding to a vanishing loopweight ($\beta = 0$), we find determinant and pfaffian expressions for thesecorrelators. We extract the conformal weights of the underlying conformalfields and find $\Delta = -\frac18$, $0$, $-\frac3{32}$, $\frac38$, $1$,$\tfrac \theta \pi (1+\tfrac{2\theta}\pi)$, where $\theta$ encodes the weightof one class of loops for the correlator of type f. These results are obtainedby analysing the asymptotics of the exact expressions, and by using theCardy-Peschel formula in the case where $x$ and $y$ are set to the corners. Fortype b, we find a $\log|x-y|$ dependence from the asymptotics, and a $\ln (\lnn)$ term in the corner free energy. This is consistent with the interpretationof the boundary condition of type b as the insertion of a logarithmic fieldbelonging to a rank two Jordan cell.For the other values of $\beta = 2 \cos \lambda$, we use the hypothesis ofconformal invariance to predict the conformal weights and find $\Delta =\Delta_{1,2}$, $\Delta_{1,3}$, $\Delta_{0,\frac12}$, $\Delta_{1,0}$,$\Delta_{1,-1}$ and $\Delta_{\frac{2\theta}\lambda+1,\frac{2\theta}\lambda+1}$,extending the results of critical dense polymers. With the results for type f,we reproduce a Coulomb gas prediction for the valence bond entanglement entropyof Jacobsen and Saleur