Scaling Properties of Current Fluctuations in Periodic TASEP

We study current fluctuations in the Totally Asymmetric Simple Exclusion Process (TASEP) on a ring with $N$ sites and $p$ particles. By introducing a deformation parameter $γ$, we analyze the tilted operator that governs the statistics of the time-integrated current. Employing the coordinate Bethe ansatz, we derive implicit expressions for the scaled cumulant generating function (SCGF), i.e. the largest eigenvalue, and the spectral gap, both in terms of Bethe roots. Their asymptotic behaviour is characterized by using the geometric structure of Cassini oval. In the thermodynamic limit at fixed particle density, we identify a dynamical phase transition separating fluctuation regimes. For $γ>0$, the SCGF exhibits ballistic growth with system size, $λ_1 \sim N$. In contrast, for $γ<0$, the SCGF converges to $-1$ as $N\to\infty$. This transition is reflected in the spectral gap, which controls the system's relaxation timescale. For $γ>0$, the gap closes at polynomial speed, $Δ\sim N^{-1}$, consistent with rapid relaxation with enhanced current. For $γ<0$, the gap vanishes exponentially, $Δ\sim \exp(-cN)$, signaling metastability with diminished current. Our non-perturbative results provide insights into large deviations and the relaxation dynamics in driven particle systems.