Directed polymers with smooth finite-correlation-range disorder beyond the quadratic (Larkin) truncation: controlled corrections, the free-energy distribution, and the crossover to asymptotic scaling

A directed polymer is an elastic string advected through a quenched random potential; its universal large-scale characteristics are the wandering exponent $ζ$, governing the growth of the typical transverse displacement with the polymer length, and the free-energy fluctuation exponent $ω$. Parisi (1990) considered the very-long-range limit in which the transverse covariance of the Gaussian disorder is Taylor-truncated at quadratic order, $C(x) ≃ a/λ² − gx² + O(x⁴/λ²)$, making the replicated problem exactly solvable: the free-energy fluctuations grow with exponent $ω = 2$ and the wandering with $ζ = 3/2$ ($⟨R_T²⟩ ∝ L³$), independent of dimension. The robustness of these results to the neglected higher-order terms is now understood. The exponent question is settled in a precise two-regime sense: the quadratic-truncation exponents are exact at all scales for the idealized random-force ensemble, but for any physical disorder with finite transverse correlation range they hold only below the Larkin length and cross over to the short-range universality class ($ζ = 2/3$, $ω = 1/3$) beyond it; the replica-limit singularities of the exact solution are likewise traced to the non-spectral (unphysical) character of any truncated covariance rather than to the replica method itself. What remains open is the systematic beyond-quadratic theory for a finite-width smooth correlator, whose central open object is the full free-energy distribution function (stated to be unsolved in the 2010 literature and not resolved since), together with the controlled higher-order corrections it entails and the description of the Larkin-to-asymptotic crossover across the Larkin length that such a theory must encompass.

Background

A directed polymer in a random medium (DPRM) is an elastic string stretched along a preferred direction (here of length $L$) through a quenched random potential $V(φ, x)$; its partition function sums over all transverse paths $φ(x)$. Because the disorder is quenched, the object of interest is the disorder-averaged free energy $⟨−T ln Z⟩$ and, more finely, the full probability distribution of the free energy $F = −T ln Z$. Two universal exponents organize the large-scale behavior in (1+1) dimensions: the wandering exponent $ζ$, fixing the growth of the typical transverse displacement as $L^ζ$ (equivalently $⟨φ²(L)⟩ ∝ L^{2ζ}$), and the free-energy fluctuation exponent $ω$, fixing the sample-to-sample spread of $F$ as $L^ω$; for the standard polymer classes they are tied by the scaling relation $2ζ = 1 + ω$. Since $⟨ln Z⟩$ is not directly computable, the standard analytic tool is the replica method: one computes the disorder average of $Z^n$ for positive integer $n$, analytically continues to non-integer $n$, and takes $n → 0$ for the averaged free energy, or integrates over complex $n$ to reconstruct the distribution of $F$ from its moments (the integer moments of $Z$ form a bilateral Laplace transform of that distribution). For $δ$-correlated (white-in-space) disorder the (1+1)-dimensional problem belongs to the Kardar-Parisi-Zhang (KPZ) universality class — the generic scaling class for interfaces growing under short-range noise, named after the stochastic growth equation the polymer maps to — with $ζ = 2/3$ and $ω = 1/3$, and its exact free-energy statistics are known from the replica Bethe ansatz (an exact-solution technique that maps the replicated polymer to a quantum many-body system solved by Bethe's wavefunction ansatz; see the account in Dotsenko et al. 2010). For smooth disorder correlated over a finite transverse distance, no exact solution is known. For such a finite-width covariance the replicated analysis has been carried out in two ways (Dotsenko et al. 2010): by expanding the correlator, and by first averaging over the random potential $V(φ, x)$ through its covariance and expanding the resulting interaction between the replicated bosons only afterwards. Parisi (1990) introduced and exactly solved the very-long-range limit: a Gaussian disorder, $δ$-correlated along the string, whose transverse covariance is Taylor-truncated at quadratic order, $C(x) ≃ a/λ² − gx² + O(x⁴/λ²)$ with $λ$ the large correlation scale. The replicated Hamiltonian is then harmonic — a system of $n$ coupled oscillators — and the analysis yields free-energy fluctuations growing as $L²$ ($ω = 2$) and wandering $⟨R_T²⟩ ∝ L³$ ($ζ = 3/2$), independent of dimension and temperature. The same fixed point arises from Larkin's idea of linearizing the random potential in the displacement, $V(φ, x) ≈ V_0(x) + f(x)φ$, and from the harmonic approximation to the correlator, $U_p(φ − φ') = U_0 − u(φ − φ')²/2$; these two constructions coincide in their short-scale behavior but are distinct approximations with different ranges of validity; the later literature loosely groups both under the names "Larkin" and "random-force", but in this problem "random-force model" denotes the shifted random-force ensemble rather than the harmonic-correlator approximation (Gorokhov and Blatter 1999; Dotsenko et al. 2010). The harmonic replica solution also exhibits a characteristic replica-limit singularity — a Gaussian coefficient diverging as the replica index is continued to zero — whose meaning became a point of contention. Two structural facts delimit the validity of the quadratic description. First, a Gaussian disorder measure is well-defined only if the covariance has a non-negative Fourier spectrum (is spectral); a finite Taylor truncation such as the quadratic one violates this at large wavevectors, so the truncated model is intrinsically unphysical outside a bounded domain (Dotsenko et al. 2010). Second, the string's wandering grows without bound, so it eventually exceeds any finite correlation width: beyond the Larkin length $L_c = (c²ξ²/u)^{1/3}$ (with $c$ the elasticity, $u$ the curvature of the correlator and $ξ$ its width) the polymer feels the decay of the true correlator and crosses over to the short-range universality class, $ζ = 2/3$ and $ω = 1/3$ (Gorokhov and Blatter 1999; Korshunov and Dotsenko 1998). The quadratic regime is thus a short- or intermediate-scale window — one that grows without bound as the correlation scale $λ → ∞$, which makes the very-long-range limit self-consistent as a limiting theory. The subsequent literature therefore treats two exactly understood limits — the idealized quadratic model, with exact moments and an exact free-energy distribution (Gorokhov and Blatter 1999; Dotsenko et al. 2010), and the asymptotic short-range problem — and a physically central, analytically open interpolation between them for smooth finite-width correlators. The broader DPRM literature (reviewed in Halpin-Healy and Zhang 1995) and subsequent crossover work (Korshunov, Geshkenbein and Blatter 2013; Caballero et al. 2022) approach that interpolation with scaling arguments, perturbation theory and numerics, but the finite-width-covariance problem has, in the words of Dotsenko et al. 2010, "remained unsolved so far".

Problem Statement

Consider the (1+1)-dimensional directed polymer — an elastic string $φ(x)$ on $[0, L]$ with elastic energy density $c(∂_xφ)²/2$ and one end pinned at the origin — in a Gaussian quenched potential $V(φ, x)$ whose statistics are

$⟨V(φ, x) V(φ', x')⟩ = δ(x − x') U(φ − φ'),$

where the transverse covariance $U(φ)$ is smooth, physical (spectral, i.e. with a non-negative Fourier transform), and of finite correlation width $ξ$, with maximum $U_0 = U(0)$ and curvature $u = −U''(0) > 0$. The free energy is $F = −T ln Z$. The free-energy distribution has two standard conventions: the fixed-endpoint distribution $P_{L,y}(F)$ with $φ(L) = y$, and the free-endpoint distribution $P_L(F)$; the two are related, in the quadratic limit, by a Gaussian convolution with the random shift (see Dotsenko et al. 2010). This is the physical regularization of the very-long-range regime in which all relevant displacement scales are small compared with $ξ$, so that $U(φ) ≈ U_0 − (u/2)φ² + O(φ⁴/ξ²)$ and the quadratic (harmonic, Larkin-type) truncation applies. The quadratic truncation exists in two distinct forms, which must not be conflated: the (shifted) random-force models, which are exactly solvable at all scales in the idealized ensemble, with free-energy fluctuations growing as $L²$ ($ω = 2$) and wandering $⟨φ²⟩ ∝ L³$ ($ζ = 3/2$); and the harmonic-correlator approximation $U_p(φ) = U_0 − uφ²/2$, which reproduces the same short-scale behavior but is valid only below the Larkin length $L_c = (c²ξ²/u)^{1/3}$ — the scale beyond which the typical transverse displacement exceeds the correlation width $ξ$ and the polymer feels the decay of the true correlator — while beyond $L_c$ its free-energy distribution develops an unphysical negative tail and its second cumulant turns negative. The short-range problem is likewise exactly characterized asymptotically ($ζ = 2/3$, $ω = 1/3$).

The open problem — in the words of Dotsenko et al. 2010, "the problem with a finite-width correlator U(φ) of the random potential has remained unsolved so far" — is the controlled beyond-quadratic theory of the polymer with this smooth finite-width disorder. It comprises three sub-questions: (a) the free-energy distribution function $P(F)$ for a finite-width smooth correlator $U(φ)$ beyond the quadratic truncation; (b) the beyond-quadratic corrections at short scales $L ≲ L_c$, going beyond the published second-order results (Dotsenko et al. 2010; Korshunov, Geshkenbein and Blatter 2013); and (c) the behavior across and beyond the Larkin length $L_c$, where the quadratic description fails and the polymer crosses over toward the short-range asymptotics. The short-scale corrections (b) are the sub-part of the crossover description (c) on scales below $L_c$.

References

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