Rigorous justification of the replica coordinate Bethe ansatz for the log-gamma polymer: Bethe-string norms, completeness, and analytic continuation in replica number

The replica coordinate Bethe ansatz is a physics technique for computing exact distribution functions of directed polymer models in the KPZ universality class: the integer moments of the partition function are mapped to an n-boson quantum problem solved by the coordinate Bethe ansatz, the result is analytically continued to complex replica number, and a formal saddle-point analysis of the sum over Bethe 'string' eigenstates yields a Fredholm determinant for the generating function of the free energy. For the log-gamma polymer this program produced the fixed-endpoint solution of Thiery and Le Doussal and, for the point-to-half-line geometry, the GUE-to-GOE free-energy crossover derived by Grange — but every step rests on unproven ingredients: the conjectured norm of Bethe string states (a Gaudin–Korepin-type formula in a non-standard weighted scalar product), the completeness of the string eigenstates in the thermodynamic limit, a Mellin–Barnes analytic continuation in the replica number necessitated by the divergence of moments of order n greater than gamma, and uncontrolled string-expansion and saddle-point asymptotics. The problem is to prove, or refute and correct, these ingredients and to show rigorously that the Fredholm determinant indeed captures the full large-time generating function. The same observables are already known rigorously by unrelated Whittaker-function methods, which validate the conjectured answers but not the Bethe-ansatz derivation itself.

Background

A directed polymer on the square lattice is a nearest-neighbour up-right path that accumulates a product of independent random Boltzmann weights attached to the visited sites; its partition function $Z$ (the sum of the path weights) is a random variable whose logarithm, the free energy, is the central observable. In one space dimension such models lie in the Kardar–Parisi–Zhang (KPZ) universality class of random growth: at large time $t$ the free energy fluctuates on the scale $t^{1/3}$, with a universal limiting one-point distribution given by a Tracy–Widom law — the GUE flavor (fluctuations of the largest eigenvalue of large random Hermitian matrices) for point-to-point ('droplet') geometries, and the GOE flavor (real symmetric matrices) for point-to-line ('flat') geometries.

The log-gamma polymer, introduced by Seppäläinen, is the canonical exactly solvable lattice polymer: the weights are inverse-gamma distributed, $P_\gamma(\omega)d\omega=\Gamma(\gamma)^{-1}\omega^{-1-\gamma}e^{-1/\omega}d\omega$ with parameter $\gamma>0$, and its fluctuation exponents and Tracy–Widom limits are rigorously known.

The replica Bethe ansatz is a physics technique for computing the full probability law of $Z$. The $n$-th integer moment $\overline{Z^n}$ (the overbar denotes averaging over the random weights) is interpreted as the wave function of an $n$-boson quantum problem: for the log-gamma polymer, the transfer matrix acting on the moments is a lattice analogue of the Lieb–Liniger Hamiltonian (the one-dimensional Bose gas with delta-function interactions) with effective coupling $\bar c=4/(\gamma-1)$. The coordinate Bethe ansatz diagonalizes such operators: eigenfunctions are superpositions of plane waves whose quasi-momenta ('rapidities') satisfy algebraic Bethe equations. This problem is of attractive type, and the contributing eigenstates organize into strings — complexes of $m$ rapidities lying regularly spaced along a segment of the complex plane, interpreted as bound-state clusters. Two structural inputs are then required: the norm of each Bethe state with respect to the relevant (here non-standard, weight-dependent) scalar product, in order to decompose the initial condition and perform the sum over eigenstates; and the completeness of the string eigenstates. For the Lieb–Liniger model with real rapidities the norm is given by the Gaudin–Korepin determinant formula (Korepin 1982), and for string states of the attractive Bose gas by the heuristic formula of Calabrese and Caux (2007). Thiery and Le Doussal (2014) conjectured the generalization of these formulas to the log-gamma polymer replica transfer matrix — a norm formula for arbitrary Bethe states (their Eq. (38)) and its thermodynamic-limit form for string states (their Eq. (52), reproduced as Eq. (27) of Grange (2017)) — assuming the string system to be complete.

A second obstruction is specific to the log-gamma polymer: because the inverse-gamma weight has a fat tail, the integer moments $\overline{Z^n}$ diverge for $n>\gamma$, so the law of $Z$ cannot be reconstructed from its integer moments. The replica Bethe ansatz instead analytically continues the moment formula to complex replica number $m$ through a Mellin–Barnes integral on a contour $C=-a+i\mathbb R$ with $a>0$ — a prescription whose validity is assumed, not proven. Finally, the sum over string configurations is resummed into a Fredholm determinant (the determinant of the identity plus an explicit integral operator, the standard representation of such limit laws), and a saddle-point analysis at large time extracts the scaling limits. For the point-to-half-line geometry, where the partition function $Z_w(x,t)=\sum_{y\in\mathbb{Z}_-}e^{wy}\sum_{\phi:(y,0)\to(x,t)}\prod_{(x',t')\in\phi}\omega_{x',t'}$ sums the weights of all up-right paths of $t$ steps ending at $(x,t)$ over starting points $y$ on the negative half-line, each weighted by $e^{wy}$ with slope $w$, this program yields a crossover between the GUE and GOE Tracy–Widom laws in the rescaled slope $\tilde w$ (Grange 2017), matching the droplet-to-flat crossover of the KPZ equation derived by the same techniques (Le Doussal 2014).

The same observables have been obtained rigorously by entirely different methods based on the geometric Robinson–Schensted–Knuth correspondence (a birational lifting of a classical combinatorial bijection) and Whittaker functions (special functions of representation-theoretic origin): Borodin–Corwin–Remenik (2013) proved the Fredholm determinant identity for the point-to-point log-gamma free energy at all times; Bisi–Zygouras (2019) computed Laplace transforms for point-to-line and point-to-half-line geometries; Barraquand–Corwin–Dimitrov (2021) proved GUE Tracy–Widom fluctuations for general parameters and slopes. These rigorous results validate the conjectured answers, but they justify no step of the replica Bethe ansatz derivation itself: the string norms, the completeness of the string eigenstates, the analytic continuation in replica number, and the string-expansion/saddle-point asymptotics all remain unproven.

Problem Statement

The problem is to give a rigorous justification of the replica coordinate Bethe ansatz derivation of the generating function of the log-gamma polymer free energy — as performed for the fixed-endpoint geometry by Thiery and Le Doussal (2014) and extended to the point-to-half-line geometry with boundary slope $w$ by Grange (2017) — by proving, or refuting and correcting, each of its conjectural ingredients. These four items are the load-bearing steps of one and the same derivation; the overall objective is a single complete rigorous derivation of the large-time generating function by the replica coordinate Bethe ansatz route, items 1-3 being prerequisites of item 4.

1. (String norms.) Prove, or disprove and correct, the conjectured norm of the Bethe string states of the log-gamma polymer replica transfer matrix in its weighted scalar product. In the thermodynamic limit of a periodic system of $L$ sites, for a system of $n_s$ strings with lengths $m_j$ and real centers $k_j$, the string rapidities are $t_{j,a}=ik_j/2+\bar c(m_j+1-2a)/4$, $a=1,\dots,m_j$, with $\bar c=4/(\gamma-1)$, and the conjectured norm (Eq. (27) of Grange (2017); Eq. (52) of Thiery-Le Doussal 2014) is

$$\|\mu\|^{2}=\Big(\sum_{j=1}^{n_s}m_j\Big)!\,L^{n_s}\prod_{1\le i<j\le n_s}\frac{4(k_i-k_j)^2+\bar c^{2}(m_i+m_j)^2}{4(k_i-k_j)^2+\bar c^{2}(m_i-m_j)^2}\prod_{j=1}^{n_s}\frac{m_j}{\bar c^{m_j-1}}\Big(\sum_{a=1}^{m_j}\frac{1}{1-t_{j,a}^2}\Big)\prod_{b=1}^{m_j}(1-t_{j,b}^2)$$

together with its finite-size version (Eq. (38) of Thiery-Le Doussal 2014). Any proof must recover the previously checked special cases (single-string norms) and reduce to the Gaudin-Korepin determinant in the continuum Lieb-Liniger limit.

2. (Completeness.) Prove that the Bethe string eigenstates used in the expansion form a complete basis, in the thermodynamic limit $L\to\infty$, of the relevant Hilbert space (the $n$-boson space on which the replica transfer matrix acts, in the weighted scalar product of item 1), so that the sum over eigenstates performed in the derivation is legitimate.

3. (Analytic continuation.) Justify the Mellin-Barnes prescription in the replica number: starting from the exact Bethe-ansatz formulas for the integer moments $\overline{Z_w(x,t)^n}$, which exist only for $n<\gamma$, prove that the continuation to complex replica order $m$ on the contour $C=-a+i\mathbb R$ and the resulting contour integral indeed reconstruct the true Laplace transform of the partition function.

4. (Asymptotics.) With the above ingredients established, rigorously control the resummation over all string configurations and the saddle-point approximation, and prove that the resulting Fredholm determinant captures the full large-time generating function of the free energy — including the GUE-to-GOE crossover in the rescaled slope — and establish its relation to the Fredholm determinants obtained from rigorous Whittaker-function methods, namely the equivalence of the large-contour Bethe-ansatz formula (the replica determinant on large open contours) with the small-contour formula of those rigorous methods (the geometric-RSK/Whittaker-function determinant).

As a secondary direction, the method can also be extended to further discrete directed polymer models (e.g., the beta and strict-weak lattice polymers) and to finite-time corrections as further tests of universality; this is a natural continuation of the same program rather than an additional requirement for completing the problem.

References

1. Pascal Grange (2017). Crossover in the log-gamma polymer from the replica coordinate Bethe Ansatz. arXiv:1707.03521. https://arxiv.org/abs/1707.03521
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