Attempted Bethe ansatz solution for one-dimensional directed polymers in random media

Boris Klumov; Victor Dotsenko

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Attempted Bethe ansatz solution for one-dimensional directed polymers in random media

Victor Dotsenko, Boris Klumov

Statistical Mechanics · Submitted 7 Jan 2010

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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arXiv ID1001.1059 (version 1)

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CitationBoris Klumov, Victor Dotsenko. "Attempted Bethe ansatz solution for one-dimensional directed polymers in random media." arXiv:1001.1059 (2010). https://pith.science/paper/IC7N4XUH

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.92 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.29 · compares with prior results 0.06 · mentions code or data availability 0.01.

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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.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	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.	contribution 0.67	0.92	0.05	asserted 1.00
s1	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.	method 0.55	0.96	0.06	asserted 1.00
s2	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.	limitation 0.72	0.86	0.10	asserted 1.00
s3	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.	result 0.92	0.93	0.74	asserted 1.00

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Original abstract

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.

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Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

It is shown that in the thermodynamic limit the problem is reduced to the Fredholm determinant with the Airy kernel yielding the universal Tracy-Widom distribution, which is known to describe the statistical properties of the Gaussian unitary ensemble as well as many other statistical systems.

Replica Bethe ansatz derivation of the Tracy-Widom distribution of the free energy fluctuations in one-dimensional directed polymers · 0.81

Papers citing it · 2 sentences without a confident label

Compares with it 1 paper

This equation agrees with the one in [ref] , although our derivation was made simpler by using results from algebraic Bethe ansatz.

Free-energy distribution of the directed polymer at high temperature · Introduction · 1.00

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