Let $G$ be an $n \times n$ grid graph. Is there anything known about the asymptotic growth rate of the number of self avoiding paths from $(0,0)$ to $(n,b)$ (from the lower left corner to some arbitrary point on the right hand side)? 


For example, is there anything known about the $b$ that maximizes this number? A limiting probability distribution on $[0,1]$ for the value of $b$ (if we pick a self avoiding walk from $(0,0)$ to $x = n$ uniformly at random)?


Some similar questions go unanswered:


[Any approximation algorithms for self-avoiding walks?](https://mathoverflow.net/questions/76857/any-approximation-algorithms-for-self-avoiding-walks)


[How does the number of self-avoiding paths between two points scale, in a square/cubic lattice?](https://mathoverflow.net/questions/108364/how-does-the-number-of-self-avoiding-paths-between-two-points-scale-in-a-square)


Thank you!



Number of self avoiding paths on a grid graph?
**UPD:** the answer below is in fact completely wrong - it deals with counting walks $\gamma$ weighted by $\mu^{-\text{length}(\gamma)}$. It is clear that without restricting or penalizing for the lengths of the walks, the number of walks will grow exponentially with the *volume* (i. e., as $\eta^{N^2}$ for some $\eta>1$). In fact, in [this paper](http://www.numdam.org/article/AIHPB_2014__50_2_315_0.pdf) it is shown that a walk chosen uniformly from this measure (or even from a measure $\hat{\mu}^{-\text{length}(\gamma)}$ with any $\hat{\mu}<\mu$) is space-filling in the scaling limit. It is also conjectured that its scaling limit is SLE$\_8$, which suggests that it is in the same universality class as the perimeter of the [Uniform Spanning tree](https://arxiv.org/abs/math/0112234). If we buy that, then the number of walks should behave similarly to the number of spanning trees (although with a different $\eta$), which is given by the determinant of the graph Laplacian. The asymptotics of the latter in rectilinear domains was [studied](https://arxiv.org/abs/math-ph/0011042), although with different boundary conditions (Kenyon has Dirichlet boundary conditions and we would be interested in Dirichlet+Neumann ones). As far as the dependence on $b$ is concerned, (i. e., if we are only interested in the ratios $P'\_n(b\_1)/P'\_n(b\_2)$ with $b\_{1,2}$ in the bulk of the right side of the square), it should be captured by the $SLE\_8$ partition function, and then the same conclusion as in discussion below might hold with a different exponent (namely, $a=\frac{6-\kappa}{2\kappa}$, which for $\kappa=8$ equals $-\frac18$). But again, these conjectures are all quite far-fetched.


**Old Answer**


[Lawler, Schramm and Werner](https://arxiv.org/pdf/math/0204277.pdf) conjectured that the number $R\_n$ of self-avoiding walks in the rectangle $[-N;N]\times[0;N]$ connecting the origin to the point $(0;N)$ behaves like 
$$
R\_n\sim c\cdot N^{-2a}\mu^N, 
$$
where $a=5/8$ and $\mu$ is a lattice-dependent quantity known as the connective constant (it is [known](http://annals.math.princeton.edu/2012/175-3/p14) to be $\sqrt{2+\sqrt{2}}$ for hexagonal lattice and only numerically known for other lattices). Moreover, they conjectured that the measure should have a conformally covariant scaling limit, meaning that if $\Omega$ is another simply-connected domain and $x,y\in \partial \Omega$, with the boundary being nice (say, horizontal or vertical straight lines) near $x,y$, then the number of SAW from $a$ to $b$ in a $\delta$-mesh lattice approximation to $\Omega$ should behave like 
$$
R^\Omega\_n \sim c |\varphi'(x)|^{a}|\varphi'(y)|^{a}\delta^{2a}\mu^{\delta^{-1}},
$$
where $\varphi$ is a conformal map from $\Omega$ to the rectangle $(-1,1)\times(0,1)$ that maps $x$ to the origin and $y$ to $i$. (We can also fix $|\varphi'(x)|=1$, and then $|\varphi'(y)|$ is proportional to the normal derivative of the Poisson kernel with a pole at $x$.) A straightforward extension of their conjecture would be that the number of SAW from the origin to $(N,b)$, with a given $b$, behaves like 
$$
P'\_n\sim c(\theta)^a\cdot N^{-3a}\mu^N,
$$
where $c(\theta)$ is proportional to the normal derivative at $(1;\theta)$ of the Poisson kernel in $(0;1)^2$ with a "pole" at the origin, and we are in the regime $b/N\to \theta\in (0,1)$. (If $b\equiv N$ or $b\equiv 0$, the power law exponent should change to $-4a$.)


No-one doubts the conformal covariance ansatz of Lawler, Schramm and Werner; but it is wide open to prove it rigorously. The best you can do rigorously is, probably, 
$$
\log P'\_n\sim n\log\mu.
$$
I am not sure this is explicitly done anywhere, but the general idea is that all 'reasonable' models of planar SAW should obey this property with the same $\mu$; for instance, [here](https://www.ihes.fr/~duminil/publi/saw_lecture_notes.pdf) it is done for arbitrary paths, bridges, and closed paths.


