Matches in SemOpenAlex for { <https://semopenalex.org/work/W3106283304> ?p ?o ?g. }
- W3106283304 endingPage "1203" @default.
- W3106283304 startingPage "1149" @default.
- W3106283304 abstract "Abstract We study the solution of the Kardar–Parisi–Zhang (KPZ) equation for the stochastic growth of an interface of height h ( x , t ) on the positive half line, equivalently the free energy of the continuum directed polymer in a half space with a wall at $$x=0$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . The boundary condition $$partial _x h(x,t)|_{x=0}=A$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:msub> <mml:mrow> <mml:mi>h</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> corresponds to an attractive wall for $$A<0$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo><</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , and leads to the binding of the polymer to the wall below the critical value $$A=-1/2$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . Here we choose the initial condition h ( x , 0) to be a Brownian motion in $$x>0$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> with drift $$-(B+1/2)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mo>-</mml:mo> <mml:mo>(</mml:mo> <mml:mi>B</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . When $$A+B rightarrow -1$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>+</mml:mo> <mml:mi>B</mml:mi> <mml:mo>→</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , the solution is stationary, i.e. $$h(cdot ,t)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>(</mml:mo> <mml:mo>·</mml:mo> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> remains at all times a Brownian motion with the same drift, up to a global height shift h (0, t ). We show that the distribution of this height shift is invariant under the exchange of parameters A and B . For any $$A,B > - 1/2$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo>></mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , we provide an exact formula characterizing the distribution of h (0, t ) at any time t , using two methods: the replica Bethe ansatz and a discretization called the log-gamma polymer, for which moment formulae were obtained. We analyze its large time asymptotics for various ranges of parameters A , B . In particular, when $$(A, B) rightarrow (-1/2, -1/2)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo>)</mml:mo> <mml:mo>→</mml:mo> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , the critical stationary case, the fluctuations of the interface are governed by a universal distribution akin to the Baik–Rains distribution arising in stationary growth on the full-line. It can be expressed in terms of a simple Fredholm determinant, or equivalently in terms of the Painlevé II transcendent. This provides an analog for the KPZ equation, of some of the results recently obtained by Betea–Ferrari–Occelli in the context of stationary half-space last-passage-percolation. From universality, we expect that limiting distributions found in both models can be shown to coincide." @default.
- W3106283304 created "2020-11-23" @default.
- W3106283304 creator A5031449563 @default.
- W3106283304 creator A5054234007 @default.
- W3106283304 creator A5072799634 @default.
- W3106283304 date "2020-08-07" @default.
- W3106283304 modified "2023-09-30" @default.
- W3106283304 title "Half-Space Stationary Kardar–Parisi–Zhang Equation" @default.
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