**A**nother trip in the métro today (to work with Pierre Jacob and Lawrence Murray in a Paris Anticafé!, as the University was closed) led me to infer—warning!, this is not the exact distribution!—the distribution of *x*, namely

since a path *x* of length *l(x)* will corresponds to N draws if N-*l(x)* is an even integer *2p* and *p* undistinguishable annihilations in 4 possible directions have to be distributed over *l(x)*+1 possible locations, with Feller’s number of distinguishable distributions as a result. With a prior π(N)=1/N on N, hence on *p*, the posterior on *p* is given by

Now, given N and *x*, the probability of no annihilation on the last round is 1 when *l(x)*=N and in general

which can be integrated against the posterior. The numerical expectation is represented for a range of values of *l(x)* in the above graph. Interestingly, the posterior probability is constant for *l(x)* large and equal to 0.8125 under a flat prior over N.

Getting back to Pierre Druilhet’s approach, he sets a flat prior on the length of the path θ and from there derives that the probability of annihilation is about 3/4. However, “the uniform prior on the paths of lengths lower or equal to M” used for this derivation which gives a probability of length l proportional to 3^{l} is quite different from the distribution of l(θ) given a number of draws N. Which as shown above looks much more like a Binomial B(N,1/2).

However, being not quite certain about the reasoning involving Fieller’s trick, I ran an ABC experiment under a flat prior restricted to (*l(x)*,4*l(x)*) and got the above, where the histogram is for a posterior sample associated with *l(x)*=195 and the gold curve is the potential posterior. Since ABC is exact in this case (i.e., I only picked N’s for which l(x)=195), ABC is not to blame for the discrepancy! I asked about the distribution on Stack Exchange maths forum (and a few colleagues here as well) but got no reply so far… Here is the R code that goes with the ABC implementation:

#observation:
elo=195
#ABC version
T=1e6
el=rep(NA,T)
N=sample(elo:(4*elo),T,rep=TRUE)
for (t in 1:T){
#generate a path
paz=sample(c(-(1:2),1:2),N[t],rep=TRUE)
#eliminate U-turns
uturn=paz[-N[t]]==-paz[-1]
while (sum(uturn>0)){
uturn[-1]=uturn[-1]*(1-
uturn[-(length(paz)-1)])
uturn=c((1:(length(paz)-1))[uturn==1],
(2:length(paz))[uturn==1])
paz=paz[-uturn]
uturn=paz[-length(paz)]==-paz[-1]
}
el[t]=length(paz)}
#subsample to get exact posterior
poster=N[abs(el-elo)==0]