## Simulating the scaling exponent in Euclidean random TSP

Take N random points in the square of area N. Fix 0 < p < 1. Consider the minimum-length cycle though some pN of the points (minimized over all choices of pN points out of N). Let f(p,N) = (length of minimum cycle)/(pN) be the average edge-length in the optimal cycle. This f(p,N) is random and depends on N, but by easy theory as N \to infinity with fixed p there is a limit non-random function f(p). Now f(1) = approx 0.72 is an already-studied constant for the random Euclidean traveling salesman problem. But I would like somebody to try to obtain the whole function f(p) via simulation. In particular,

(a) theory says the limit f(0+) is strictly > 0; estimate it numerically.

(b) statistical physics suggests a scaling exponent \alpha:

#### f(p) - f(0+) scales as p^\alpha for small p

estimate \alpha.

(November 2006) I have done some crude simulations myself, but these are far too crude to help with the scaling exponent question.