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Add references: [MW67] and [MW73]

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Ian Jauslin 2016-09-11 23:55:47 +00:00
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Giuliani_Jauslin_Lieb_2015.tex
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@ -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, 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. 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. 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 $\mathbb R^2$, 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 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}]. 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}]. 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 $M_1(i_j,i_{j'})$. 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 \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 \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. \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 $1\le i\le |c|$, \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 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 \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 \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 string of vertices in common appearing in the reverse order, that is

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bibliography.BBlog.tex
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@ -27,6 +27,8 @@
\BBlogentry{Li67}{Li67}{E.H. Lieb - {\it Solution of the dimer problem by the transfer matrix method}, Journal of Mathematical Physics, Vol.~8, n.~12, p.~2339-2341, 1967, doi:{\tt\color{blue}\href{http://dx.doi.org/10.1063/1.1705163}{10.1063/1.1705163}}.} \BBlogentry{Li67}{Li67}{E.H. Lieb - {\it Solution of the dimer problem by the transfer matrix method}, Journal of Mathematical Physics, Vol.~8, n.~12, p.~2339-2341, 1967, doi:{\tt\color{blue}\href{http://dx.doi.org/10.1063/1.1705163}{10.1063/1.1705163}}.}
\BBlogentry{Li68}{Li68}{E.H. Lieb - {\it A theorem on Pfaffians}, Journal of Combinatorial Theory, Vol.~5, p.~313-319, 1968, doi:{\tt\color{blue}\href{http://dx.doi.org/10.1016/S0021-9800(68)80078-X}{10.1016/S0021-9800(68)80078-X}}.} \BBlogentry{Li68}{Li68}{E.H. Lieb - {\it A theorem on Pfaffians}, Journal of Combinatorial Theory, Vol.~5, p.~313-319, 1968, doi:{\tt\color{blue}\href{http://dx.doi.org/10.1016/S0021-9800(68)80078-X}{10.1016/S0021-9800(68)80078-X}}.}
\BBlogentry{LL93}{LL93}{E.H. Lieb, M. Loss - {\it Fluxes, Laplacians, and Kasteleyn's Theorem}, Duke Mathematical Journal, Vol.~71, n.~2, p.~337-363, 1993.} \BBlogentry{LL93}{LL93}{E.H. Lieb, M. Loss - {\it Fluxes, Laplacians, and Kasteleyn's Theorem}, Duke Mathematical Journal, Vol.~71, n.~2, p.~337-363, 1993.}
\BBlogentry{MW67}{MW67}{B.M. McCoy, T.T. Wu - {\it Theory of Toeplitz Determinants and the Spin Correlations of the Two-Dimensional Ising Model. IV}, Physical Review, Vol.~162, n.~2, p.~436-475, 1967, doi:{\tt\color{blue}\href{http://dx.doi.org/10.1103/PhysRev.162.436}{10.1103/PhysRev.162.436}}.}
\BBlogentry{MW73}{MW73}{B.M. McCoy, T.T. Wu - {\it The Two-Dimensional Ising Model}, Harvard University Press, 1973.}
\BBlogentry{PS}{PS}{H. Pinson, T. Spencer - {\it Universality and the two-dimensional Ising model}, unpublished.} \BBlogentry{PS}{PS}{H. Pinson, T. Spencer - {\it Universality and the two-dimensional Ising model}, unpublished.}
\BBlogentry{PR08}{PR08}{V.B. Priezzhev, P. Ruelle - {\it Boundary monomers in the dimer model}, Physical Review E, Vol.~77, n.~061126, 2008, doi:{\tt\color{blue}\href{http://dx.doi.org/10.1103/PhysRevE.77.061126}{10.1103/PhysRevE.77.061126}}.} \BBlogentry{PR08}{PR08}{V.B. Priezzhev, P. Ruelle - {\it Boundary monomers in the dimer model}, Physical Review E, Vol.~77, n.~061126, 2008, doi:{\tt\color{blue}\href{http://dx.doi.org/10.1103/PhysRevE.77.061126}{10.1103/PhysRevE.77.061126}}.}
\BBlogentry{Sm01}{Sm01}{S. Smirnov - {\it Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits}, Comptes Rendus de l'Acad\'emie des Sciences - Series I - Mathematics, Vol.~333, n.~3, p.~239-244, 2001, doi:{\tt\color{blue}\href{http://dx.doi.org/10.1016/S0764-4442(01)01991-7}{10.1016/S0764-4442(01)01991-7}}.} \BBlogentry{Sm01}{Sm01}{S. Smirnov - {\it Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits}, Comptes Rendus de l'Acad\'emie des Sciences - Series I - Mathematics, Vol.~333, n.~3, p.~239-244, 2001, doi:{\tt\color{blue}\href{http://dx.doi.org/10.1016/S0764-4442(01)01991-7}{10.1016/S0764-4442(01)01991-7}}.}