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Free-energy distribution of the directed polymer at high temperature
Disordered Systems and Neural Networks · Submitted 24 Feb 2010
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The exact free energy distribution is obtained for any time and compared with very recent results on growth and exclusion models.
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We study the directed polymer of length $t$ in a random potential with fixed endpoints in dimension 1+1 in the continuum and on the square lattice, by analytical and numerical methods. The universal regime of high temperature $T$ is described, upon scaling 'time' $t \sim T^5/\kappa$ and space $x = T^3/\kappa$ (with $\kappa=T$ for the discrete model) by a continuum model with $\delta$-function disorder correlation. Using the Bethe Ansatz solution for the attractive boson problem, we obtain all positive integer moments of the partition function. The lowest cumulants of the free energy are predicted at small time and found in agreement with numerics. We then obtain the exact expression at any time for the generating function of the free energy distribution, in terms of a Fredholm determinant. At large time we find that it crosses over to the Tracy Widom distribution (TW) which describes the fixed $T$ infinite $t$ limit. The exact free energy distribution is obtained for any time and compared with very recent results on growth and exclusion models.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study the directed polymer of length $t$ in a random potential with fixed endpoints in dimension 1+1 in the continuum and on the square lattice, by analytical and numerical methods. | contribution 0.94 | 0.95 | 0.03 | asserted 0.93 |
| s1 | The universal regime of high temperature $T$ is described, upon scaling 'time' $t \sim T^5/\kappa$ and space $x = T^3/\kappa$ (with $\kappa=T$ for the discrete model) by a continuum model with $\delta$-function disorder correlation. | method 0.74 | 0.66 | 0.08 | asserted 1.00 |
| s2 | Using the Bethe Ansatz solution for the attractive boson problem, we obtain all positive integer moments of the partition function. | method 0.75 | 0.94 | 0.07 | asserted 1.00 |
| s3 | The lowest cumulants of the free energy are predicted at small time and found in agreement with numerics. | result 0.99 | 0.91 | 0.09 | asserted 0.99 |
| s4 | We then obtain the exact expression at any time for the generating function of the free energy distribution, in terms of a Fredholm determinant. | result 0.90 | 0.96 | 0.30 | asserted 1.00 |
| s5 | At large time we find that it crosses over to the Tracy Widom distribution (TW) which describes the fixed $T$ infinite $t$ limit. | result 0.99 | 0.92 | 0.31 | asserted 1.00 |
| s6 | The exact free energy distribution is obtained for any time and compared with very recent results on growth and exclusion models. | result 0.82 | 0.93 | 0.18 | asserted 1.00 |
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Original abstract
We study the directed polymer of length $t$ in a random potential with fixed endpoints in dimension 1+1 in the continuum and on the square lattice, by analytical and numerical methods. The universal regime of high temperature $T$ is described, upon scaling 'time' $t \sim T^5/\kappa$ and space $x = T^3/\kappa$ (with $\kappa=T$ for the discrete model) by a continuum model with $\delta$-function disorder correlation. Using the Bethe Ansatz solution for the attractive boson problem, we obtain all positive integer moments of the partition function. The lowest cumulants of the free energy are predicted at small time and found in agreement with numerics. We then obtain the exact expression at any time for the generating function of the free energy distribution, in terms of a Fredholm determinant. At large time we find that it crosses over to the Tracy Widom distribution (TW) which describes the fixed $T$ infinite $t$ limit. The exact free energy distribution is obtained for any time and compared with very recent results on growth and exclusion models.
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How the literature uses it
21 arXiv papers cite this in 42 sentences
Papers citing it · 27 sentences without a confident label
Uses its method or tool 3 papers
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We can now invoke the expression for the Cauchy determinant and the consequent identities used in previous works [ref] to decouple the strings in the inverse norm factor [equation] where a set of $n_s$ real (positive) auxiliary variables has been introduced.
Tail of the two-time height distribution for KPZ growth in one dimension · Calculation of FIXME · 1.00
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E.1, i.e. using the standard determinant double-Cauchy identity: Φ(k, m) = ∏_j=1^n_s (2 m_j) _1 ≤i, j ≤n_s[ 1/2 i (k_i-k_j) + m_i + m_j] performing the rescaling $k_j →k_j/λ$, denoting $x̃=x/λ^2$ and performing the Airy trick, i.e. the identity $e^1/3 λ^3 m^3 = ∫dy (y) e^λm y$ for $λm>0$, we obtain Z^dec_λ(n_s,s_1,s_L,s_R;x̃) = 2^n_s ∏_j=1^n_s [ ∑_m_j=1^∞∫dk_j/2 π dy_j (y_j) (-1)^m_j e^- i λm_j k_j x̃ - 4 λm_j k_j^2 + λm_j y_j ] ×_1 ≤i, j ≤n_s[ 1/2 i (k_i-k_j) + λm_i + λm_j] ×∑_ m^L + m^1 + m^R = m ∏_j=1^n_s S^w_L,a_L_m_j^L,m_j,k_j/λ S^w_R,a_R_m_j^R,m_j,-k_j/λ e^- λm^L_j s_L - λm^1_j s_1 - λm^R_j s_R Using standard manipulations [ref] the partition sum at fixed number of string $n_s$ can thus be expressed itself as a determinant: Z^dec_λ(n_s,s_1,s_L,s_R;x̃) = ∏_j=1^n_s ∫_v_j>0 _1 ≤i, j ≤n_s M^λ_s_1,s_L,s_R,x̃(v_i,v_j) with the Kernel: M^λ_s_1,s_L,s_R,x̃(v_i,v_j) = ∫dk/2 π dy (y + 4 k^2 +
Maximum of an Airy process plus Brownian motion and memory in KPZ growth · III.4. Large time limit and decoupling assumption: first form of the kernel · 0.99
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Here, as in [ref] , we use a different strategy by imposing to the Bethe eigenfunctions artificial boundary conditions on a line of length $L$, and studying the solutions of the associated Bethe equations in the limit $L →∞$.
Exact solution for a random walk in a time-dependent 1D random environment: the point-to-point Beta polymer · Bethe Ansatz solution of the Beta polymer · 0.94
Compares with it 3 papers
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One recovers exactly the result obtained in [ref] ($u$ there being $-f'$ here), by a completely different method.
Tail of the two-time height distribution for KPZ growth in one dimension · Constrained summations · 0.98
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In fact we have not found a continuum limit of the Beta polymer which would identify to the repulsive LL model.], while the moment problems of the Inverse-Beta polymer (see [ref] ), the log-Gamma polymer (see [ref] ) and the continuum directed polymer (see [ref] ) were similar to the attractive phase.
Exact solution for a random walk in a time-dependent 1D random environment: the point-to-point Beta polymer · Resolution of the Bethe equations in the large $L$ limit: repulsion and free particles · 0.99
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While this manuscript was in course of preparation I have learned that the exactly the same result for the system considered in this paper has been independently derived by P.Calabrese, P.Le Doussal and A.Rosso [ref] .
Bethe anzats derivation of the Tracy-Widom distribution for one-dimensional directed polymers · Introduction · 0.98
Cites it as background 5 papers
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An alternative approach to the solution of the continuum model (with KPZ universality properties related to classes of boundary conditions [ref] ), is the study of discrete models of directed polymers (see [ref] for a recent review, and [ref] for more families of models and their classification, and [ref] for rigorous solutions involving the replica approach).
Crossover in the log-gamma polymer from the replica coordinate Bethe Ansatz · Introduction and conclusions · 0.99
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At the center of this universality class lies the continuum KPZ equation [ref] , for which exact results were obtained more recently, for the three main types of initial conditions (droplet, flat, stationary) [ref] .
Tail of the two-time height distribution for KPZ growth in one dimension · Introduction · 0.97
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While this research program has been completed for continuum models, yielding insights on the KPZ universality class [ref] , an alternative approach is the study of discrete models of directed polymers (or integrable particle systems or growth models), whose solutions can yield results for the continuum model by taking a suitable limit (see [ref] for a recent review, and [ref] for more families of models and their classification, and [ref] for rigorous solutions involving the replica approach).
Log-gamma directed polymer with one free end via coordinate Bethe Ansatz · Introduction and conclusions · 0.97
What it cites · 2 sentences without a confident label
Compares with 3 papers
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Upon rescaling of $v,v'$ by $2^-1/3$ our result ([ref]) can also be rewritten as: [equation] where $B_t$ is the kernel defined in [ref] hence the results coincide.
The one-dimensional KPZ equation: an exact solution and its universality · Introduction · 0.99
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This equation agrees with the one in [ref] , although our derivation was made simpler by using results from algebraic Bethe ansatz.
Attempted Bethe ansatz solution for one-dimensional directed polymers in random media · Introduction · 1.00
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We can now compare our result with the very recent works [ref] on KPZ growth with the narrow wedge initial condition, to which our work also applies (see also [ref] ).
Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions · Introduction · 1.00
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