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Diffstat (limited to 'Giuliani_Jauslin_Lieb_2015.tex')
| -rw-r--r-- | Giuliani_Jauslin_Lieb_2015.tex | 7 |
1 files changed, 4 insertions, 3 deletions
diff --git a/Giuliani_Jauslin_Lieb_2015.tex b/Giuliani_Jauslin_Lieb_2015.tex index e539376..4744ccc 100644 --- a/Giuliani_Jauslin_Lieb_2015.tex +++ b/Giuliani_Jauslin_Lieb_2015.tex @@ -77,8 +77,9 @@ diagonalized as in [\cite{Li67}]. The partition function can then be computed by if it is a finite portion of a lattice) they decay polynomially at large distances, like $1/(distance)$, asymptotically as the size of the graph tends to infinity. See [\cite{PR08}] for a proof of this fact on the square lattice on the half-plane. +A similar analysis has been worked out in the 2D nearest neighbor Ising model for the boundary free energy, in the presence of a boundary magnetic field, and for the boundary spin-spin correlations, see [\cite{MW67}, Section 8] and [\cite{MW73}, Chapters VI and VII]. If the graph is a discrete, regular, approximation of a finite domain of $\mathbb R^2$, -the scaling limit of these correlations +the scaling limit of the boundary monomer correlations at close-packing is expected to exist and to be conformally invariant under conformal mappings of the domain, in analogy with other observables of the critical 2D Ising model and of the close-packed dimer model [\cite{Ke00}, \cite{Ke01}, \cite{Sm01}, \cite{Sm10}, \cite{CHI15}, \cite{Du11}, \cite{Du15}]. In particular, they are expected to coincide with those of complex chiral free fermions [\cite{PR08}]. @@ -180,7 +181,7 @@ M_n(i_1,\cdots,i_{2n})=\mathrm{pf}(\mathfrak M_{i_1,\ldots,i_{2n}}), where $\mathfrak M_{i_1,\ldots,i_{2n}}$ is the $2n\times 2n$ antisymmetric matrix whose $(j,j')$-th entry with $j<j'$ is $M_1(i_j,i_{j'})$. \endtheo -{\bf Remark}: Away from close packing (i.e. omitting $\ell_1=\cdots=\ell_{|g|}$ in~(\ref{eqcorrdef})), {\it the Wick rule does not hold}. This can be checked immediately by considering a graph consisting of a square with an extra edge on the diagonal. +{\bf Remark}: Away from close packing (i.e. omitting $\ell_1=\cdots=\ell_{|g|}=0$ in~(\ref{eqcorrdef})), {\it the Wick rule does not hold}. This can be checked immediately by considering a graph consisting of a square with an extra edge on the diagonal. \bigskip \indent As stated in theorem~\ref{theomain}, the edges and vertices of $g$ must be directed and labeled in a special way. In particular, the direction of the edges must satisfied a so called {\it Kasteleyn} condition, and the labeling must satisfy a {\it positivity} condition. The positivity condition ensures that the terms that appear in the Pfaffian add up constructively and reproduce the MD partition function. The Kasteleyn condition is used to prove the positivity of a graph: if such a condition holds, then it suffices to look at a single dimer covering of $g$ to prove its positivity. @@ -241,7 +242,7 @@ show how to reduce general graphs to graphs with a boundary circuit. \indent Given an edge $\{v_i,v_{i+1}\}$ for $1\le i\le |c|$, the edge is said to be \defd{forwards} if $v_i\succ v_{i+1}$ and \defd{backwards} if $v_i\prec v_{i+1}$; and similarly for -$\{v_{|c|},v_1\}$. A circuit $c$ is said to be \defd{oddly-directed} if it contains an {\it odd} number of {\it backwards} edges and \defd{evenly-directed} if it contains an {\it even} number of {\it backwards} edges. In addition a circuit will said to be \defd{good} if it is {\it oddly-directed} and encloses an {\it even} number of vertices, or it is {\it evenly-directed} and encloses an {\it odd} number of vertices. +$\{v_{|c|},v_1\}$. A circuit $c$ is said to be \defd{oddly-directed} if it contains an {\it odd} number of {\it backwards} edges and \defd{evenly-directed} if it contains an {\it even} number of {\it backwards} edges. In addition a circuit is said to be \defd{good} if it is {\it oddly-directed} and encloses an {\it even} number of vertices, or it is {\it evenly-directed} and encloses an {\it odd} number of vertices. \indent Furthermore, given $\nu\ge 1$ and two circuits $c_1$ and $c_2$ that have a string of vertices in common appearing in the reverse order, that is |