Tail decay for the distribution of the endpoint of a directed polymer

Thomas Bothner, Karl Liechty
2013 Nonlinearity  
We obtain an asymptotic expansion for the tails of the random variable =_u∈R(A_2(u)-u^2) where A_2 is the Airy_2 process. Using the formula of Schehr Sch that connects the density function of to the Hastings-McLeod solution of the second Painlevé equation, we prove that as t→∞, P(||>t)=Ce^-4/3φ(t)t^-145/32(1+O(t^-3/4)), where φ(t)=t^3-2t^3/2+3t^3/4, and the constant C is given explicitly.
doi:10.1088/0951-7715/26/5/1449 fatcat:xnzy6kmotnafxhv6nnkgkmbwt4