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\begin{document}

\section{Lattice effects for SAW - conjectures}

Following is for the square lattice. Other lattices (triangular, hexagonal)
will require some modifications.

For $\ell \in (0,1)$ and an angle $\theta$, let $L$ be the line with 
polar angle $\theta$ passing through $(0,-\ell)$. Let $H$ be the 
half plane determined by $L$ that contains the origin. 
Let $c_N$ be the number of SAW's starting at the origin with $N$ 
steps. Let $b_N$ be the number of these walks that stay in $H$. 
So $b_N/c_N$ is the probability the $N$ step SAW stays in $H$. 
It is conjectured that 
\beann
{b_N \over c_N} \asymp N^\rho
\eeann
with $\rho=25/64$ \cite{LSW}.
We define
\beann
h(\theta,\ell)= \lim_{N \ra \infty} {b_N \over c_N} N^{-\rho}
\eeann
and then define 
\beann
h(\theta) = \int_0^1 h(\theta,\ell) d\ell
\eeann
We have averaged over vertical shifts of the line. This should be equivalent 
to averaging over sites in $H$ which are adjacent to $L$ in the sense
that there is a bond from the site to a site not in $H$, and starting 
the walk at such sites.
Since $\theta$ and $\theta+\pi$ give the same line, 
$h(\theta)$ has period $\pi$. The symmetry with respect to 
relections in the vertical axis implies $h(\theta)=h(\pi-\theta)$. 

Define $b(\theta)$ to be the density of vertical bonds along a line 
with polar angle $\theta$, i.e, the average number of vertical 
bonds that interesect the line per unit length. 
We have $b(\theta)=|\cos(\theta)|$. The density of horizontal 
lines is $b(\theta-\pi/2)$.

Take a simple curve $C$ in the half plane $\Half$ that starts and 
ends on the boundary and encloses the origin (e.g., a semicircle centered
at the origin). 
We have $\Half \setminus C = D_i \cup D_e$
where $D_i$ is the part of the half plane inside the curve and $D_e$ the 
part of the half plane outside the curve. 
We condition on the event that the SAW intersects 
this curve exactly once. 
We conjecture that the hitting density along this curve for 
$z \in C$ is proportional to 
\bea
H(D_i;0,z) H(D_e;z,\infty) \ell(z,D)
\label{hitting}
\eea
This is the density with respect to arc length.
Here $H(D;z,w)$ is the SLE partition function, and 
$\ell(z,D)$ is the lattice correction factor. It is given by 
\bea
\ell(z,D) = b(\theta) h(\theta)^2 + b(\theta-\pi/2) h(\theta-\pi/2)^2 
\label{ldef}
\eea
where $\theta=\theta(z,D)$ is the polar angle of the tangent line to 
$D$ at $z$. The two terms in $\ell(z,D)$ are the contributions from 
horizontal bonds crossing $C$ and vertical bonds crossing $C$.

Let $D$ be a simply connected domain with smooth boundary. 
Let $C_1$ and $C_2$ be two disjoint boundary arcs. 
Introduce the lattice $\delta \integers^2$. 
The total mass of the SAW's in $D$ from $C_1$ to $C_2$ is 
\bea
Z_\delta (D;C_1,C_2) = \delta^2 \sum_{z \in C_1} \sum_{v \in C_2} 
\sum_{\omega:z \ra v, \omega \subset D}
\beta^{|\omega|}
\label{domainconj}
\eea
The factor of $\delta^2$ is to normalize the sums over $z$ and $v$ 
so that they approximate integrals of $z$ and $v$ over $C_1$ and $C_2$. 
We define the sums over $z \in C_1$ and $v \in C_2$ as follows. 
(Other definitions are possible.) For $z \in C_1$, we sum over all 
sites outside of $D$ such that there is a bond which crosses $C_1$, 
and so connects $z$ to a site inside of $D$.  (Note that 
some $z \in C_1$ will have more than one such bond.) 
The sum over walks $\omega$ is over walks that only visit sites in $D$ 
except for starting at $z$ and ending at $v$.
The conjecture is that 

\beann
\lim_{\delta \ra 0} \delta^{-2 b} Z_\delta(D;C_1,C_2) 
= \int_{\partial D} \int_{\partial D} H(D;z,v) 
c(z,D) c(v,D) |dz| |dv|
\eeann
where $c(z,D) = b(\theta(z,D)) \, h(\theta(z,D))$.
Note that here we just have $h(\theta(z,D))$, not $h(\theta(z,D))^2$ as 
before.

\bigskip

Another form of the conjecture is 
$$
\lim_{\delta \rightarrow 0} \sum_{z,v \in \partial D} \delta^{2-2b}
\mu(z,v) = 
\int_{\partial D} \int_{\partial D} \nu_D(z,v) c(z) c(v) |dz| |dv|
$$
where $\mu(z,v)$ is the unnormalized SAW measure on walks from $z$ to 
$v$. $\nu_D(z,v)$ is the unnormalized SLE measure. 
This measure valued statement does not have any more content than 
the conjecture \reff{domainconj}.

\section{Computations}

There are two types of computations to be done. The first is to 
compute the function $h(\theta)$. This can be done by exact enumeration 
of walks or by a Monte Carlo simulation to compute the 
probability a SAW starting at the origin stays in the half plane. 

\bigskip

The other type of computation is to test the conjectured hitting 
density \reff{hitting} for various geometries. See figure.
Note that this only tests the first conjecture.

(a) {\bf full plane walk and circle:} Take the SAW in the full plane 
starting at the origin and a circle centered at the origin. 
Condition on crossing the circle once. The SLE partition functions 
are trivial and so the hitting density is just $\ell(\theta)$.

(b) {\bf half plane walk and semi-circle:} Take the SAW in the half plane
starting at the origin. Condition on it crossing a semicircle 
centered at the origin exactly one. 

(c) {\bf tilted slit:} For the SAW in half plane, 
take a ray starting at  $y<0$ with angle $\phi$. Condition on 
crossing this line exactly once and look at the hitting density 
along the line. Since the orientation of the lattice with respect to 
the line is constant, there is no lattice effect. So we can test the 
SLE partition function prediction directly.

(d) {\bf half plane walk and triangle:} Take the SAW in a half plane
but with the angle between the boundary of the half plane and a lattice 
direction to be $\phi$. Take an isoceles triangle with base on 
the boundary of the half plane and with two angles of $\phi$.
So one side of the triangle is parallel to a lattice direction. 
Condition on crossing the part of the triangle off the boundary of the 
half plane exactly once. 
Since the lattice is oriented differently with respect to the 
two sides, the probabilities of hitting the two sides are not equal. 

(e) {\bf full plane walk and off center circle:}
Take a full plane walk starting at the origin, but consider a 
circle which contains the origin but is not centered there. 
This is a test of radial SLE partition function predictions.

\begin{figure}[tbh]
%\includegraphics{sims.eps}
\caption{\leftskip=25 pt \rightskip= 25 pt 
Simulations to be done
}
\label{sims}
\end{figure}


The first graph shows $h(\theta)$. It is a computed by a Monte Carlo
simulation of the probability the full plane SAW stays in a half plane. 
The curve shows uses walks of length $500$. The apparent smoothness
of the curve is a reflection of the large number of samples generated. 

The second graph has two curves. The red solid line is the factor 
$\ell(\theta)$ computed using $h(\theta)$ and eq. \reff{ldef}. 
The blue dashed line is the hitting density for a full plane 
walk conditioned to cross a circle exactly once. This is a density 
on $[0,360]$, so the portion of the curve shown has area $1/4$. 
We have scaled $h(\theta)$ so that this curve also has area $1/4$. 

\begin{figure}[tbh]
%\includegraphics{h.eps}
\caption{\leftskip=25 pt \rightskip= 25 pt 
The function $h(\theta)$.
}
\end{figure}


\begin{figure}[tbh]
%\includegraphics{l.eps}
\caption{\leftskip=25 pt \rightskip= 25 pt 
The function $\ell(\theta)$ and the density for the full plane SAW conditioned
to hit a circle exactly once. 
}
\end{figure}


\end{document}
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