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1.
Phys Rev E ; 104(2-1): 024502, 2021 Aug.
Artigo em Inglês | MEDLINE | ID: mdl-34525573

RESUMO

We present an exact solution for the height distribution of the KPZ equation at any time t in a half space with flat initial condition. This is equivalent to obtaining the free-energy distribution of a polymer of length t pinned at a wall at a single point. In the large t limit a binding transition takes place upon increasing the attractiveness of the wall. Around the critical point we find the same statistics as in the Baik-Ben-Arous-Péché transition for outlier eigenvalues in random matrix theory. In the bound phase, we obtain the exact measure for the endpoint and the midpoint of the polymer at large time. We also unveil curious identities in distribution between partition functions in half-space and certain partition functions in full space for Brownian-type initial condition.

2.
J Stat Phys ; 181(4): 1149-1203, 2020.
Artigo em Inglês | MEDLINE | ID: mdl-33087988

RESUMO

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 . The boundary condition ∂ x h ( x , t ) | x = 0 = A corresponds to an attractive wall for A < 0 , and leads to the binding of the polymer to the wall below the critical value A = - 1 / 2 . Here we choose the initial condition h(x, 0) to be a Brownian motion in x > 0 with drift - ( B + 1 / 2 ) . When A + B → - 1 , the solution is stationary, i.e. h ( · , t ) 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 , 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 ) → ( - 1 / 2 , - 1 / 2 ) , 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.

3.
Phys Rev E ; 101(4-1): 040101, 2020 Apr.
Artigo em Inglês | MEDLINE | ID: mdl-32422817

RESUMO

We study the Kardar-Parisi-Zhang (KPZ) growth equation in one dimension with a noise variance c(t) depending on time. We find that for c(t)∝t^{-α} there is a transition at α=1/2. When α>1/2, the solution saturates at large times towards a nonuniversal limiting distribution. When α<1/2 the fluctuation field is governed by scaling exponents depending on α and the limiting statistics are similar to the case when c(t) is constant. We investigate this problem using different methods: (1) Elementary changes of variables mapping the time-dependent case to variants of the KPZ equation with constant variance of the noise but in a deformed potential. (2) An exactly solvable discretization, the log-gamma polymer model. (3) Numerical simulations.

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