Limit shapes of large skew Young tableaux and a modification of the TASEP process

The paper’s Theorem 2 states that for a ring of length L with N stones, the topological entropy equals log(sin(πN/L)/sin(π/L)) and that the maximal-entropy state distribution is determinantal with kernel the projection onto any N consecutive Fourier harmonics. The proof in Section 4.2 diagonalizes the one-step adjacency via exterior powers of the right-shift (Fourier modes) and identifies the spectral radius as the maximum of sums of N Lth roots of unity, attained by consecutive blocks, and then derives the determinantal law via the projection kernel K = ∑ v_r v_r^* (cf. formula (22)). The chain’s period is L, as noted later when discussing path counts W_n and the Poissonized limit, which is consistent with the existence of L cyclic components of the MME. All of these match the candidate solution, which uses the same Fourier/second-quantization diagonalization and determinantal projection, with a slightly more explicit extremal bound (Fejér/Dirichlet-kernel) for the sum-of-roots-of-unity maximum. Minor differences: the paper explicitly treats the even-N phase/gauge tweak to ensure nonnegativity/positivity, while the candidate’s Doob-transform formula for the transition kernel implicitly assumes a positive eigenvector; this is easily fixed by the paper’s gauge choice. Overall, both arguments align on substance and conclusions (entropy and determinantal MME) and use substantially the same proof strategy. Key points are in 1.2 and Theorem 2’s statement, the exterior-power diagonalization and entropy computation in 4.2, the determinantal stationary law in (22), and the discussion of period L in Lemma 2’s proof.