Tilting Billingsley's model toward a giant prime: two phase transitions

Source abstract

Billingsley's theorem states that the normalized logarithms of the prime factors of a uniform random integer in $[1,x]$ converge to the Poisson--Dirichlet law $PD(1)$, while weighting an integer $n$ by the generalized divisor function $d_θ(n)$ gives $PD(θ)$. We ask what remains of this picture when the integer is also rewarded according to its largest prime factor $P^+(n)$. For fixed $θ,β>0$ and $γ\ge0$, we sample $N_x=n\le x$ with probability proportional to $d_θ(n)\exp\{βH_γ(n)\}$, where $H_γ(n)=(\log P^+(n))^γ$ for $γ>0$, while $H_0(1)=0$ and $H_0(n)=1$ for $n\ge2$. Two phase transitions occur. At $γ=0$ the $PD(θ)$ partition survives, whereas every fixed $γ>0$ forces one prime to carry asymptotically all logarithmic mass. The second transition, at $γ=1$, concerns the cofactor $R_x=N_x/P^+(N_x)$. With $a_x=βγ(\log x)^{γ-1}$, its law is asymptotic in total variation to $Q_{a_x}(m)=d_θ(m)m^{-1-a_x}/ζ(1+a_x)^θ$. Thus, for $0 1$, $R_x=1$ with high probability. We also determine the joint limits involving $N_x/x$ and the normalizing constants, which include the classical Alladi--Erdős asymptotic as a special case.