Spectrum of the totally asymmetric simple exclusion process on a periodic lattice -- first excited states

We consider the spectrum of the totally asymmetric simple exclusion process on a periodic lattice of $L$ sites. The first eigenstates have an eigenvalue with real part scaling as $L^{-3/2}$ for large $L$ with finite density of particles. Bethe ansatz shows that these eigenstates are characterized by four finite sets of positive half-integers, or equivalently by two integer partitions. Each corresponding eigenvalue is found to be equal to the value at its saddle point of a function indexed by the four sets. Our derivation of the large $L$ asymptotics relies on a version of the Euler-Maclaurin formula with square root singularities at both ends of the summation range.