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!

up vote 9 down vote accepted

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 $$ R_n\sim c\cdot N^{-2a}\mu^N, $$ where $a=5/8$ and $\mu$ is a lattice-dependent quantity known as the connective constant (it is known 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 it is done for arbitrary paths, bridges, and closed paths.

  • Great answer, thanks! Do you think you could say a little more explicitly what $c(b)$ is? Which Poisson kernel? I'm also a little confused by the notation: $P_n'(b)$ refers to the number of SAW from $(0,0)$ to $(n,b)$? – Lorenzo Oct 16 at 21:03
  • So the conjecture is that probability distribution is going to be proportional to $c(\theta)^a$ (in the regime $\theta \in (0,1)$)? It would be helpful for me if you could write the conjectured probability distribution a little more explicitly, since I'm not too familiar with this stuff. I will try to read the papers you suggest, thank you! – Lorenzo Oct 16 at 21:11
  • yes, the notation was perhaps unclear, I edited the text. The Poisson kernel refers to the unique non-negative harmonic function in the domain that extends continuously to all points of the boundary except for $x$; the normalization may be whatever it is - the asymptotics includes some unknown, lattice-dependent multiplicative constant anyways. For the square, I guess, you can write the Poisson kernel explicitly using elliptic functions. – Kostya_I Oct 16 at 22:32
  • Are there any families of graphs or lattices where an exact count of the number of self avoiding walks between points is known? – Lorenzo Oct 17 at 16:37

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 $\mu=\lim\limits_{n\in \mathbb{N}} c_{n}^{\frac{1}{n}}=\lim\limits_{n\in 2\mathbb{N}} p_{n}^{\frac{1}{n}}$. According to Madras' paper here an upper bound for $\mathbb{Z}^2$ is given by $p_n \leq Cn^{−1/2}\mu^n$ 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' $p_n \leq Cn^{−1/2}\mu^n$ 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).

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