Diffusion of a set of random walkers in Euclidean media. First passage times

When a large number N of independent random walkers diffuse on a d-dimensio nal Euclidean substrate, what is the expectation value [t(1,N)] of the time : spent by the first random walker to cross a given distance r from the sta rting place?: We here explore the relationship between this quantity and th e number of different sites visited by N random walkers all starring from t he same origin. This leads us to conjecture that [t(1,N)] approximate to (r (2)/4D ln N)[1+ Sigma(n=1)(infinity) (ln N)(-n) Sigma(m=0)(n) a(m)((n)) (ln ln N)(m)] for d greater than or equal to 2, large N and r much greater tha n ln N, where a(m)((n)) are constants (some of which we estimate numericall y) and D is the diffusion constant. Pie find this conjecture to be compatib le with computer simulations.