A stationary model of non-intersecting directed polymers

Abstract

Abstract We consider the partition function <?CDATA $Z_{\ell}(\vec x,0\vert \vec y,t)$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mrow> <mml:mi>ℓ</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mover> <mml:mi>x</mml:mi> <mml:mo>⃗</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo fence="false" stretchy="false">|</mml:mo> <mml:mover> <mml:mi>y</mml:mi> <mml:mo>⃗</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> of <?CDATA $\ell$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>ℓ</mml:mi> </mml:math> non-intersecting continuous directed polymers of length t in dimension <?CDATA $1+1$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:math> , in a white noise environment, starting from positions <?CDATA $\vec x$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover> <mml:mi>x</mml:mi> <mml:mo>⃗</mml:mo> </mml:mover> </mml:math> and terminating at positions <?CDATA $\vec y$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover> <mml:mi>y</mml:mi> <mml:mo>⃗</mml:mo> </mml:mover> </mml:math> . When <?CDATA $\ell = 1$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>ℓ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:math> , it is well known that for fixed x , the field <?CDATA $\log Z_1(x,0\vert y,t)$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>log</mml:mi> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>|</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> solves the Kardar–Parisi–Zhang equation and admits the Brownian motion as a stationary measure. In particular, as t goes to infinity, <?CDATA $Z_1(x,0\vert y,t)/Z_1(x,0\vert 0,t) $?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo fence="false" stretchy="false">|</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo fence="false" stretchy="false">|</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> converges to the exponential of a Brownian motion B ( y ). In this article, we show an analogue of this result for any <?CDATA $\ell$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>ℓ</mml:mi> </mml:math> . We show that <?CDATA $Z_{\ell}(\vec x,0\vert \vec y,t)/Z_{\ell}(\vec x,0\vert \vec 0,t) $?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mrow> <mml:mi>ℓ</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mover> <mml:mi>x</mml:mi> <mml:mo>⃗</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo fence="false" stretchy="false">|</mml:mo> <mml:mover> <mml:mi>y</mml:mi> <mml:mo>⃗</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>Z</mml:mi> <mml:mrow> <mml:mi>ℓ</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mover> <mml:mi>x</mml:mi> <mml:mo>⃗</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo fence="false" stretchy="false">|</mml:mo> <mml:mover> <mml:mn>0</mml:mn> <mml:mo>⃗</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> converges as t goes to infinity to an explicit functional <?CDATA $Z_{\ell}^\mathrm{\,stat}(\vec y)$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi>Z</mml:mi> <mml:mrow> <mml:mi>ℓ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">t</mml:mi> <mml:mi mathvariant="normal">a</mml:mi> <mml:mi mathvariant="normal">t</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mover> <mml:mi>y</mml:mi> <mml:mo>⃗</mml:mo> </mml:mover> <mml:mo stretchy="false">)</mml:mo> </mml:math> of <?CDATA $\ell$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>ℓ</mml:mi> </mml:math> independent Brownian motions. This functional <?CDATA $Z_{\ell}^\mathrm{\,stat}(\vec y)$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi>Z</mml:mi> <mml:mrow> <mml:mi>ℓ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">t</mml:mi> <mml:mi mathvariant="normal">a</mml:mi> <mml:mi mathvariant="normal">t</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mover> <mml:mi>y</mml:mi> <mml:mo>⃗</mml:mo> </mml:mover> <mml:mo stretchy="false">)</mml:mo> </mml:math> admits a simple description as the partition sum for <?CDATA $\ell$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>ℓ</mml:mi> </mml:math> non-intersecting semi-discrete polymers on <?CDATA $\ell$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>ℓ</mml:mi> </mml:math> lines. We discuss applications to the endpoints and midpoints distribution for long non-crossing polymers and derive explicit formula in the case of two polymers. To obtain these results, we show that the stationary measure of the O’Connell–Warren multilayer stochastic heat equation is given by a collection of independent Brownian motions. This in turn is shown via analogous results in a discrete setup for the so-called log-gamma polymer and exploit the connection between non-intersecting log-gamma polymers and the geometric Robinson–Schensted–Knuth correspondence found in Corwin-O’Connell-Seppäläinen-Zygouras (2014 Duke Math. J. 163 513–63).

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