Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the  half-line

Krajenbrink, Alexandre; Doussal, Pierre Le

doi:10.21468/scipostphys.8.3.035

Cited by 27 publications

(70 citation statements)

References 79 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the domain b −1/2 we show that the present results agree with the solution obtained in Ref. [82]. Although they are formally valid at all times, they will be analyzed here only in the limit of large time.…”

Section: Introduction and Aim Of The Papersupporting

confidence: 90%

“…In Section 3.4 we show that this generating function can be written as a Fredholm Pfaffian for all time, and obtain an exact expression for its kernel for all b and t. In Section 3.5 we show that our result is consistent with the one of Ref. [82] and discuss the Mellin-Barnes summation procedure in Section (3.6). From then on we only focus on the large time limit.…”

Section: Overview Of the Paper And Main Resultssupporting

confidence: 70%

“…With the help of the recent results in [82], we show in Section 3.7 the emergence of: the GSE Tracy-Widom distribution for b > −1/2 and the GOE Tracy-Widom distribution for b = −1/2. In Section 3.8 we show the emergence of the Gaussian distribution for b < −1/2.…”

Section: Overview Of the Paper And Main Resultsmentioning

confidence: 90%

“…was proved to be the one-dimensional GSE kernel [76,77]. Here F 4 is the CDF of the GSE-TW distribution (same convention as in [76,82]). • Critical boundary regime: b = −1/2.…”

Section: Rescaled Free-energy Kpz Height and Definition Of The Genermentioning

confidence: 99%

“…The present paper is a companion paper of Ref. [82]. Here we do not make use of the aforementioned duality and study the droplet IC for any value of b directly from the replica Bethe ansatz in the half-space.…”

Section: Introduction and Aim Of The Papermentioning

confidence: 99%

See 4 more Smart Citations

Delta-Bose gas on a half-line and the Kardar–Parisi–Zhang equation: boundary bound states and unbinding transitions

Nardis¹,

Krajenbrink²,

Doussal³

et al. 2020

J. Stat. Mech.

Self Cite

View full text Add to dashboard Cite

We revisit the Lieb-Liniger model for n bosons in one dimension with attractive delta interaction in a half-space R + with diagonal boundary conditions. This model is integrable for arbitrary value of b ∈ R, the interaction parameter with the boundary. We show that its spectrum exhibits a sequence of transitions, as b is decreased from the hard-wall case b = +∞, with successive appearance of boundary bound states (or boundary modes) which we fully characterize. We apply these results to study the Kardar-Parisi-Zhang equation for the growth of a one-dimensional interface of height h(x, t), on the half-space with boundary condition ∂ x h(x, t)| x=0 = b and droplet initial condition at the wall. We obtain explicit expressions, valid at all time t and arbitrary b, for the integer exponential (one-point) moments of the KPZ height field e nh(0,t) . From these moments we extract the large time limit of the probability distribution function (PDF) of the scaled KPZ height function. It exhibits a phase transition, related to the unbinding to the wall of the equivalent directed polymer problem, with two phases: (i) unbound for b > − 1 2 where the PDF is given by the GSE Tracy-Widom distribution (ii) bound for b < − 1 2 , where the PDF is a Gaussian. At the critical point b = − 1 2 , the PDF is given by the GOE Tracy-Widom distribution.Overview: KPZ in a half-space.There has been much recent progress in physics and mathematics in the study of the 1D (KPZ) universality class, thanks to the discovery of exact solutions and the development of powerful methods to address stochastic integrability and integrable probability. The KPZ class includes a host of models [1]: discrete versions of stochastic interface growth such as the PNG growth model [2][3][4], exclusion processes such as the TASEP, the ASEP, the q-TASEP and other variants [5][6][7][8][9][10][11][12][13], discrete (i.e. square lattice) [3,[14][15][16][17][18][19][20][21][22] or semi-discrete [23-25] models of directed polymers (DP) at zero and finite temperature, random walks in time dependent random media [26,27], dimer models, random tilings, random permutations [28], correlation function in quantum condensates [29][30][31], and more. At the center of this class lies the continuum KPZ equation [32], see equation (3), which describes the stochastic growth of a continuum interface, and its equivalent formulation in terms of continuum directed polymers (DP) [33], via the Cole-Hopf mapping onto the stochastic heat equation (SHE). Recently exact solutions have also been obtained for the KPZ equation at all times for various initial conditions [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50]. This was achieved by two different routes. First by studying scaling limits of solvable discrete models, which allowed for rigorous treatments. The second, pioneered by Kardar [51], is non-rigorous, but leads to a more direct solution: it starts from the DP formulation, uses the replica method together with a mapping to the attractive delta-Bose gas (LL model), whi...

show abstract

Section: Introduction and Aim Of The Papersupporting

confidence: 90%

Section: Overview Of the Paper And Main Resultssupporting

confidence: 70%

Section: Overview Of the Paper And Main Resultsmentioning

confidence: 90%

“…was proved to be the one-dimensional GSE kernel [76,77]. Here F 4 is the CDF of the GSE-TW distribution (same convention as in [76,82]). • Critical boundary regime: b = −1/2.…”

Section: Rescaled Free-energy Kpz Height and Definition Of The Genermentioning

confidence: 99%

Section: Introduction and Aim Of The Papermentioning

confidence: 99%

See 3 more Smart Citations

Delta-Bose gas on a half-line and the Kardar–Parisi–Zhang equation: boundary bound states and unbinding transitions

Nardis¹,

Krajenbrink²,

Doussal³

et al. 2020

J. Stat. Mech.

Self Cite

View full text Add to dashboard Cite

show abstract

Stationary Half-Space Last Passage Percolation

Betea

Ferrari

Occelli

2020

Commun. Math. Phys.

View full text Add to dashboard Cite

In this paper we study stationary last passage percolation (LPP) in half-space geometry. We determine the limiting distribution of the last passage time in a critical window close to the origin. The result is a new two-parameter family of distributions: one parameter for the strength of the diagonal bounding the half-space (strength of the source at the origin in the equivalent TASEP language) and the other for the distance of the point of observation from the origin. It should be compared with the one-parameter family giving the Baik-Rains distributions for full-space geometry. We finally show that far enough away from the characteristic line, our distributions indeed converge to the Baik-Rains family. We derive our results using a related integrable model having Pfaffian structure together with careful analytic continuation and steepest descent analysis.is obtained as a limit of some specific two-sided random initial condition 2 . Further results for random but not necessarily stationary initial conditions are also known [29,31,38,68].For further details and recent developments around the KPZ universality class, see also the following surveys and lecture notes: [25,30,34,41,67,69,76,79].In this paper we consider a stationary model in half-space, where the latter means having a height function h(x, t) defined only on x ∈ N (or x ∈ R + ). Our model, called stationary half-space last passage percolation (LPP), is defined in Section 2.1. In this geometry there are considerably fewer results compared to the case of full-space geometry. Of course, one has to prescribe the dynamics at site x = 0. If the influence on the height function of the growth mechanism at x = 0 is very strong, then close to the origin one will essentially see fluctuations induced by it, and since the dynamics in KPZ models has to be local (in space but also in time), one will observe Gaussian fluctuations. If the influence of the origin is small, then it will not be seen in the asymptotic behavior. Between the two situations there is typically a critical value where a third different distribution function is observed. Furthermore, under a critical scaling, one obtains a family of distributions interpolating between the two extremes. For some versions of half-space LPP and related stochastic growth models (with non-random initial conditions) this has indeed been proven: one has a transition of the one-point distribution from Gaussian to GOE Tracy-Widom at the critical value, and GSE Tracy-Widom distribution [9,13,57,72] 3 Furthermore, the limit process under critical scaling around the origin is also analyzed and the transition processes have been characterized [3,4,16,72].However, the limiting distribution of the stationary LPP in half-space remained unresolved. In this paper we close this gap: in Theorem 2.3 we determine the distribution function of the stationary LPP for the finite size system and in Theorem 2.6 we determine the large time limiting distribution under critical scaling. Finally, we show that in a special limit one recovers the ...

show abstract

Half-Space Stationary Kardar–Parisi–Zhang Equation

2020

Self Cite

View full text Add to dashboard Cite

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

show abstract

Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line

Cited by 27 publications

References 79 publications

Delta-Bose gas on a half-line and the Kardar–Parisi–Zhang equation: boundary bound states and unbinding transitions

Delta-Bose gas on a half-line and the Kardar–Parisi–Zhang equation: boundary bound states and unbinding transitions

Stationary Half-Space Last Passage Percolation

Half-Space Stationary Kardar–Parisi–Zhang Equation

Contact Info

Product

Resources

About