On the importance sampling of self-avoiding walks

TL;DR

This work analyzes the variance of sequential importance sampling for self-avoiding walks that cross a $k\times k$ square. Extending Knuth’s square-crossing sampler to NES-walks (including South steps) yields a second-moment growth $\mathbb{E}(X_{k,\ell}^2)\sim\alpha_k\rho_k^{-\ell}$ with a unique real pole $\rho_k$ and leads to a quasi-exponential growth of the relative variance, specifically $\operatorname{Var}(X_{k,\ell})/\mathbb{E}(X_{k,\ell})^2 \sim \tfrac{3}{2}\left(\frac{2^{k+1}}{(k+1)^2}\right)^{\ell}$ for width $\ell$ within certain regimes. The average walk length is $\sim k\ell/3$, so the variance is exponential in the typical length, mirroring Rosenbluth-type behavior for unconfined SAWs. The paper also establishes exact generating functions for the second moment in fixed-height strips, asymptotics via kernel methods, and a broader discussion of implications for unconfined walks, kinetic sampling, and trapping. Together, these results provide rare, rigorous insight into the variance of importance-sampling estimators for SAWs and clarify when such estimators become exponentially inefficient.

Abstract

In a 1976 paper published in Science, Knuth presented an algorithm to sample (non-uniform) self-avoiding walks crossing a square of side k. From this sample, he constructed an estimator for the number of such walks. The quality of this estimator is directly related to the (relative) variance of a certain random variable X_k. From his experiments, Knuth suspected that this variance was extremely large (so that the estimator would not be very efficient). But how large? For the analogous Rosenbluth algorithm, which samples unconfined self-avoiding walks of length n, the variance of the corresponding estimator is believed to be exponential in n. A few years ago, Bassetti and Diaconis showed that, for a sampler à la Knuth, that generates walks crossing a k\times k square and consisting of North and East steps, the relative variance is only O(\sqrt k). In this note we take one step further and show that, for walks consisting of North, South and East steps, the relative variance jumps to 2^{k(k+1)}/(k+1)^{2k}. This is quasi-exponential in the average length of the walks, which is of order k^2. We also obtain partial results for general self-avoiding walks crossing a square, suggesting that the relative variance could be exponential in k^2 (which is again the average length of these walks). Knuth's algorithm is a basic example of a widely used technique called sequential importance sampling. The present paper, following Bassetti and Diaconis' paper, is one of very few examples where the variance of the estimator can be found.

Figures (10)

• Figure 1: The 12 self-avoiding walks crossing the $2\times 2$ square. For each of them, we give the sequence $1/p_1, 1/p_2, \ldots$ where $p_i$ is the probability of the $i$th step. The probability of the walk is thus the reciprocal of the product of the terms in the list. Two walks have probability $1/8$, six have probability $1/12$, and four have probability $1/16$. Two walks that differ by a diagonal symmetry have the same probability.
• Figure 2: Left: A SAW crossing the $10\times 10$ square. The thick steps have probability 1. That is, each of them is the only eligible step at the time when it is taken. Right: A SAW crossing the $100\times 100$ square, obtained via Knuth's algorithm.
• Figure 3: The nine N E S-walks crossing a $2\times 2$ square, with the reciprocals of the probabilities of their steps.
• Figure 4: Recursive construction of bounded N E S-walks.
• Figure 5: Left: A plot of the modulus of $S$, showing the cut on the interval $[-1,-1/9]$. Right: The function $R(s)$.

Theorems & Definitions (15)

• Lemma 1
• proof
• Proposition 2
• Lemma 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof
• Lemma 6