mjhjmtu

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?

How does the number of self-avoiding paths between two points scale, in a square/cubic lattice?

Thank you!

Lawler, Schramm and Werner 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
$$
R_nsim ccdot N^-2amu^N,
$$
where $a=5/8$ and $mu$ is a lattice-dependent quantity known as the connective constant (it is known to be $sqrt2+sqrt2$ 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 $x,yin partial Omega$, 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
$$
R^Omega_n sim c |varphi'(x)|^a|varphi'(y)|^adelta^2amu^delta^-1,
$$
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
$$
P'_nsim c(theta)^acdot N^-3amu^N,
$$
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 $b/Nto thetain (0,1)$. (If $bequiv N$ or $bequiv 0$, 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,
$$
log P'_nsim nlogmu.
$$
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 it is done for arbitrary paths, bridges, and closed paths.

The problem you are interested in is related to enumerating self-avoiding walks in a graph (one can consider self-avoiding polygons, which are closed SAWs, also) and is a difficult problem computationally.

For an infinite graph, the asymptotic behavior of the number of length $n$ self-avoiding walks (or self-avoiding polygons) is denoted $c_n$ ($p_n$ resp.) and depends on the structure of one's graph namely on the connective constant $mu=limlimits_nin mathbbN c_n^frac1n=limlimits_nin 2mathbbN p_n^frac1n$.
According to Madras' paper here an upper bound for $mathbbZ^2$ is given by $p_n leq Cn^−1/2mu^n$ with a better estimate anticipated (see @Kostya_I's answer).

Edit: @Kostya_I's comments on conformal mappings supercedes most of the information here. The best estimate that was found after looking through a few papers improves Madras' $p_n leq Cn^−1/2mu^n$ mentioned above to $p_n ≤ n^−3/2+o(1)mu^n$ by A. Hammond here (for a set of even $n$ of full density).