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arXiv:2608.03730

A Spectral Route to Directed-Polymer Glasses

Sen Mu, Abbas Ali Saberi, Mehran Kardar A

Published Aug 4, 2026• 0 citations
arXivHTMLPDF

Abstract

A finite density of mutually avoiding directed polymers in a quenched random medium is a minimal model of glassy line matter. The dilute theory, solved by replica Bethe ansatz, predicts an interaction free energy proportional to $ρ^2$ and disorder cumulants with distinct power-law dependences on the density $ρ$, but direct numerical tests have been hindered by the combinatorially large many-polymer transfer matrix. We recast the problem as filling logarithmic eigenvalues of a single-polymer transfer-matrix product, obtaining the quenched free energy, its cumulants, and a disorder-induced linear spectral edge consistent with the replica prediction.

DOI
10.48550/arxiv.2608.03730

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A finite density of mutually avoiding directed polymers in a quenched random medium is a minimal model of glassy line matter. The dilute theory, solved by replica Bethe ansatz, predicts an interaction free energy proportional to ρ2ρ^2ρ2 and disorder cumulants with distinct power-law dependences on the density ρρρ, but direct numerical tests have been hindered by the combinatorially large many-polymer transfer matrix. We recast the problem as filling logarithmic eigenvalues of a single-polymer transfer-matrix product, obtaining the quenched free energy, its cumulants, and a disorder-induced linear spectral edge consistent with the replica prediction.
We study the statistical properties of one-dimensional directed polymers in a short-range random potential by mapping the replicated problem to a many-body quantum boson system with attractive interactions. We find the full set of eigenvalues and eigenfunctions of the many-body system and perform the summation over the entire spectrum of excited states. The analytic continuation of the obtained exact expression for the replica partition function from integer to non-integer replica parameter N turns out to be ambiguous. Performing the analytic continuation simply by assuming that the parameter N can take arbitrary complex values, and going to the thermodynamic limit of the original directed polymer problem, we obtain the explicit universal expression for the probability distribution function of free energy fluctuations.
The techniques developed by Edwards and Anderson (1975) for the spin-glass problem are here applied to the bond-diluted quenched Ising model (S= 1 / 2 ). The n to 0 trick is used to express exactly the free energy in terms of a generalized Hamiltonian representing n different but coupled replicas. The only approximation of the theory consists of decoupling these replicas in a way that reduces the free energy of the diluted system to that of a pure system with renormalized, concentration-dependent parameters. To first order in the concentration of missing bonds the author obtains all the results previously derived by Harris (1974) using a cumulant expansion and which are exact in this limit. More generally, the theory allows a discussion of the concentration dependence of the critical temperature; in particular, the values deduced for the critical concentration agree rather well with those obtained exactly or through series expansions. The behaviour of the system near the transition is also studied and the critical exponents are found to be Fisher-renormalized.
We analyze, via Imry-Ma scaling arguments, the strong disorder phases that exist in low dimensions at all temperatures for directed polymers and interfaces in random media. For the uncorrelated Gaussian disorder, we obtain that the optimal strategy for the polymer in dimension 1+d with 0<d<2 involves at the same time (i) a confinement in a favorable tube of radius R(S) approximately L (nu(S) ) with nu(S) =1/(4-d)<1/2 (ii) a superdiffusive behavior R approximately Lnu with nu=(3-d)/(4-d)>1/2 for the wandering of the best favorable tube available. The corresponding free energy then scales as F approximately Lomega with omega=2nu-1 and the left tail of the probability distribution involves a stretched exponential of exponent eta=(4-d)/2. These results generalize the well known exact exponents nu=2/3, omega=1/3, and eta=3/2 in d=1, where the subleading transverse length R(S) approximately L(1/3) is known as the typical distance between two replicas in the Bethe ansatz wave function. We then extend our approach to correlated disorder in transverse directions with exponent alpha and/or to manifolds in dimension D+d= d(t) with 0<D<2. The strategy of being both confined and superdiffusive is still optimal for decaying correlations ( alpha<0 ), whereas it is not for growing correlations ( alpha>0 ). In particular, for an interface of dimension ( d(t) -1) in a space of total dimension 5/3< d(t) <3 with random-bond disorder, our approach yields the confinement exponent nu(S) =( d(t) -1)(3- d(t) )/(5 d(t) -7). Finally, we study the exponents in the presence of an algebraic tail 1/ V1+micro in the disorder distribution, and obtain various regimes in the (micro,d) plane.
We study the continuum field theory for an ensemble of directed polymers r_i (t) in 1+d' dimensions that live in a medium with quenched point disorder and interact via short-ranged pair forces g Ψ(r_i - r_j). In the strong-disorder (or low-temperature) regime, such forces are found to be relevant in any dimension d' below the upper critical dimension for a single line. Attractive forces generate a bound state with localization length ξ_\perp \sim |g|^{-ν_\perp}; repulsive forces lead to mutual avoidance with a pair distribution function P(r_i - r_j) \sim |r_i - r_j|^θreminiscent of interacting fermions. In the experimentally important dimension d' = 2, we obtain ν_\perp \approx 0.8 and θ\approx 2.4 .
The 'cavity-mean-field' approximation developed for the random transverse field Ising model on the Cayley tree (Ioffe and Mézard 2010 Phys. Rev. Lett. 105 037001) has been found to reproduce the known exact result for the surface magnetization in d = 1 (Dimitrova and Mézard 2011 J. Stat. Mech. P01020). In the present paper, we propose to extend these ideas in finite dimensions d > 1 via a non-linear transfer approach for the surface magnetization. In the disordered phase, the linearization of the transfer equations corresponds to the transfer matrix for a directed polymer in a random medium of transverse dimension D = d − 1, in agreement with the leading order perturbative scaling analysis (Monthus and Garel 2011 arXiv: 1110.3145 ). We present numerical results of the non-linear transfer approach in dimensions d = 2 and 3. In both cases, we find that the critical point is governed by infinite disorder scaling. In particular, exactly at criticality, the one-point surface magnetization scales as , where ω c ( d ) coincides with the droplet exponent ω DP ( D = d − 1) of the corresponding directed polymer model, with ω c ( d = 2) = 1/3 and . The distribution P ( v ) of the positive random variable v of order O(1) presents a power-law singularity near the origin with a ( d = 2, 3) > 0, so all moments of the surface magnetization are governed by the same power-law decay with x s = ω c (1 + a ) independently of the order k .