The symmetry of a knot, s, is the number of moves to the right minus the number of moves to the left. The knot class contains knots with s=0 if is even; otherwise, the optimal symmetry is -1 or 1. We estimate, for a given class, the symmetry probability density , asymptotic in h.
The move composition of a knot sequence, and hence its symmetry, corresponds to the coordinates of the analogous random walk. We begin by rewriting the evolution equations (1) for the persistent random walk given that the fraction steps are along the c-axis,
These give rise to a walk in which the fraction of steps tends toward , while remaining symmetric between and .
Since we are only interested in the and step composition, we project the 2-dimensional walk on to the perpendicular to the c-axis, say, the x-axis, reducing the problem to a 1-dimensional persistent random walk of steps of or . In this simplified walk, a step to the left is followed with probability by another step to the left and by a step to the right; a step to the right is similarly biased toward the left. The resulting evolution equations are written
where is the conditional probability that the walker is at x on the nth step, having just taken a step along the positive x-axis, and v is conditioned on a step along the negative x-axis.
By the central limit theorem, the distribution of the symmetry of the projected random walk
where if the ith step is and if the ith step is , approaches a gaussian for large m. Accordingly, we desire the projected walk's mean and variance.
The evolution equations (15) are symmetric about 0 apart from the initial step, which is always . Observing the possible paths taken in the first few steps of the unprojected walk, it is evident that the first moment satisfies
and, accordingly, the mean is
In what follows, we make use of the local correlation function, . It may be observed that
where denotes the average over . By considering the general average as successive averages over , , etc., we have
The second moment may be expressed in terms of the local correlation function as
Separating the sum into i=j and terms, we have
substituting in (20), it follows that
Since the mean is always bounded by [-1,0], we approximate the variance as
Together with (18), the above expression specifies the distribution of the terminal coordinates of the walk and, consequently, the symmetry of the tie.