Directed random polymers via nested contour integrals

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• 作者：Alexei Borodin^a ; ^b ; ^{borodin@math.mit.edu" class="auth_mail" title="E-mail the corresponding author} ; Alexey Bufetov^a ; ^c ; ^{alexey.bufetov@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Ivan Corwin^d ; ^e ; ^{ivan.corwin@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Kardar&ndash ; Parisi&ndash ; Zhang ; Directed polymers ; Bethe ansatz ; Lieb&ndash ; Liniger model ; Delta Bose gas
• 刊名：Annals of Physics
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：368
• 期：Complete
• 页码：191-247
• 全文大小：716 K

文摘

We study the partition function of two versions of the continuum directed polymer in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616000373&_mathId=si48.gif&_user=111111111&_pii=S0003491616000373&_rdoc=1&_issn=00034916&md5=7d956e30ab3da1cd72b76a690340d884" title="Click to view the MathML source">1+1class="mathContainer hidden">class="mathCode">$1+1$ dimension. In the full-space version, the polymer starts at the origin and is free to move transversally in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616000373&_mathId=si49.gif&_user=111111111&_pii=S0003491616000373&_rdoc=1&_issn=00034916&md5=9edf1bfbbf92e30b749f62a5007872c5" title="Click to view the MathML source">Rclass="mathContainer hidden">class="mathCode">$\mathbb{R}$, and in the half-space version, the polymer starts at the origin but is reflected at the origin and stays in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616000373&_mathId=si50.gif&_user=111111111&_pii=S0003491616000373&_rdoc=1&_issn=00034916&md5=ea4d82490f282628ce5e293d2bc38969" title="Click to view the MathML source">R₋class="mathContainer hidden">class="mathCode">${\mathbb{R}}_{-}$. The partition functions solve the stochastic heat equation in full-space or half-space with mixed boundary condition at the origin; or equivalently the free energy satisfies the Kardar–Parisi–Zhang equation.

We derive exact formulas for the Laplace transforms of the partition functions. In the full-space this is expressed as a Fredholm determinant while in the half-space this is expressed as a Fredholm Pfaffian. Taking long-time asymptotics we show that the limiting free energy fluctuations scale with exponent class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616000373&_mathId=si51.gif&_user=111111111&_pii=S0003491616000373&_rdoc=1&_issn=00034916&md5=521a45f89d28cc9dbda4543ded01136e" title="Click to view the MathML source">1/3class="mathContainer hidden">class="mathCode">$1/3$ and are given by the GUE and GSE Tracy–Widom distributions. These formulas come from summing divergent moment generating functions, hence are not mathematically justified.

The primary purpose of this work is to present a mathematical perspective on the polymer replica method which is used to derive these results. In contrast to other replica method work, we do not appeal directly to the Bethe ansatz for the Lieb–Liniger model but rather utilize nested contour integral formulas for moments as well as their residue expansions.