Open problems on the spiked spectral boundary of the ASEP: Bethe-root clustering, restricted-cycle random graphs, and Bethe-ansatz completeness

The Asymmetric Simple Exclusion Process (ASEP) is a paradigmatic model of one-dimensional driven transport. For finite chains, the eigenvalues of its Markov generator accumulate, after rescaling, in a region of the complex plane whose boundary carries a sharp combinatorial structure: $L$ spikes for periodic boundaries and $L+1$ for open boundaries, where $L$ is the chain length. Nakerst, Prosen and Haque [Phys. Rev. E 110, 014110 (2024)] showed that in the non-interacting limit these spikes follow from sums of single-particle eigenvalues lying on circles or ellipses, that for periodic totally asymmetric ASEP at any interaction strength they arise, as demonstrated numerically, from clustering of coordinate-Bethe-ansatz roots around $L$ single-particle positions, and that they persist in random-graph ensembles whose cycle lengths are all multiples of $L$. They left open a four-part program, recorded here as one problem: extend the clustering mechanism to the partially asymmetric and open-boundary cases; derive the spectral-boundary law for restricted-cycle random-graph ensembles as an extension of the universal hypotrochoidic law of random matrix theory; and classify which other classical or quantum many-body generators exhibit spiked spectral boundaries.

Background

The Asymmetric Simple Exclusion Process (ASEP) is a stochastic lattice gas central to non-equilibrium statistical mechanics: particles hop right with rate $p$ and left with rate $q$, subject to hard-core exclusion (at most one particle per site). The totally asymmetric case ($q=0$) is called TASEP. On a ring of $L$ sites with $N$ particles (periodic boundary conditions, pbc), or on an open segment where particles enter and leave at boundary rates (open boundary conditions, obc), the ASEP defines a finite continuous-time Markov chain whose generator is a large matrix of dimension $\binom{L}{N}$ (pbc) or $2^L$ (obc). Writing the generator as a non-Hermitian Hamiltonian $H$ acting on fermionic occupation states, the standard ASEP corresponds to an interacting model with interaction strength $U=1$, while $U=0$ is a quadratic (free-fermion) model; Nakerst, Prosen and Haque introduced this tunable $U$ to interpolate between the two. The many-body eigenvalues of $H$, rescaled by $1/L$, accumulate in a bounded region of the complex plane, and the boundary of this region — the spectral boundary — carries a striking number of sharp outward spikes: $L$ spikes for pbc and $L+1$ for obc. In the free case this is elementary: the single-particle eigenvalues lie on a circle (TASEP) or ellipse (ASEP), many-body eigenvalues are sums of $N$ of them, and the most-rotated sums stick out as spikes. For interacting TASEP on a ring the spectrum is described by the coordinate Bethe ansatz, the standard tool of integrable systems in which eigenfunctions are superpositions of plane waves whose momenta (Bethe roots) solve coupled algebraic equations and eigenvalues are sums of one-particle functions of the roots. For $q=0$ and general $U$, the Bethe equations collapse so that every root satisfies the same degree-$L$ polynomial $P(Z)=(U+Z)^{L-N}(U-Z)^{N}-Y,$ with a right-hand side $Y$ independent of the root index; the Bethe roots then form $L$ clusters around the single-particle positions, with fitted widths shrinking as $L^{-1/2}$ at half filling (demonstrated numerically), which is the source of the spikes. For $q\neq 0$ the equations couple all roots through index-dependent right-hand sides and no analogous reduction is known; the obc Bethe equations (solved by de Gier and Essler and successors) are functional equations of considerably higher complexity. A complementary viewpoint is combinatorial: configurations are vertices of a directed many-body graph whose edges are allowed transitions, and all cycle lengths in this graph are divisible by $L$ (pbc) or $L+1$ (obc); random directed-graph ensembles sharing this restricted cycle structure also display spiked spectral densities. The relevant universality result is the hypotrochoidic law of Aceituno, Rogers and Schomerus: for non-Hermitian random matrices whose correlations are dominated by a single cycle length (equivalently, random graphs with dominant cycles), the spectral boundary follows a universal hypotrochoid, the roulette curve traced by a point on a circle rolling inside a fixed circle. That law, however, allows cycles of many lengths, and the restricted ensembles constructed for the ASEP fall outside its hypotheses. Finally, spectral decompositions of ASEP used in exact finite-volume fluctuation formulas rely on completeness of the Bethe ansatz, i.e. the statement that the Bethe construction yields all eigenstates; completeness for the standard periodic ASEP was proved by Brattain, Do and Saenz using algebro-geometric techniques, but the interacting general-$U$ Bethe equations introduced by Nakerst, Prosen and Haque are supported only by numerical evidence of completeness.

Problem Statement

The problem, as posed in the conclusion of Nakerst, Prosen and Haque (Phys. Rev. E 110, 014110 (2024), arXiv:2402.00662), is to complete the mechanistic and universal account of the spiked spectral boundary of ASEP-type many-body Markov generators. The common scientific objective is to explain the $L$ (pbc) or $L+1$ (obc) spikes of the spectral boundary of the ASEP generator across all asymmetries and boundary conditions, and to ground that explanation in the graph-theoretic structure that makes it universal. The requested results are: each of (i)–(iv) is an independently sufficient contribution; the task is resolved by a rigorous, checkable resolution of any one part, and a partial resolution of every part does not suffice.

(i) Determine whether the Bethe-root clustering mechanism established for periodic TASEP persists for the partially asymmetric process with $p\neq 1$ and $q\neq 0$ (the standard ASEP, $U=1$); if it does, characterize how the interaction strength $U$ influences Bethe-root clustering and the emergence of the spiky spectral boundary in the $q\neq 0$ family, where the Bethe equations no longer reduce to a single polynomial, and if it does not, identify and justify the mechanism that actually produces the spiky spectral boundary in that family;

(ii) Determine whether the $L+1$ spikes of the open-boundary spectral boundary arise from an analogous clustering of Bethe roots; if they do, adapt the coordinate Bethe ansatz or another method to produce an explicit clustering-based mechanism for the open interacting case, whose Bethe equations are more complex, and if they do not, identify and justify the mechanism that actually produces them;

(iii) Derive the spectral-boundary law for the restricted-cycle random-graph ensemble in which all cycle lengths are divisible by $L$ (or $L+1$), extending the universal hypotrochoidic law for random matrices with cyclic correlations to this ensemble class;

(iv) Identify which other classical or quantum many-body generators, beyond the ASEP, produce spiked spectral boundaries purely from the cycle or trace structure of their many-body graphs.

References

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