Riemann surfaces for KPZ with periodic boundaries

Prolhac, Sylvain

doi:10.21468/scipostphys.8.1.008

Cited by 23 publications

(56 citation statements)

References 574 publications

(1,195 reference statements)

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“…As mentioned before, if we consider the one-point function and step initial condition, Quastel-Remenik's result is Corollary 1.3 for u = U KPZ . 7 In the context of the periodic TASEP, Prolhac [37] noticed that ∂ xx det(I − K z ) is "a reminiscent of soliton solutions for the KP equation". Corollary 1.3 establishes a precise connection.…”

Section: Integrable Differential Equationsmentioning

confidence: 99%

“…To evaluate this integral asymptotically using the method of steepest-descent, it turned out that we need to consider an analytic continuation of polylog functions on a Riemann surface and the main contribution comes from a boundary point on a new sheet of the Riemann surface. Recently motivated by the same function F (x; τ, γ), Prolhac [37] studied Riemann surfaces associated to general polylog functions. In this paper, we carry out the analytic continuation directly for polylog functions of positive half integer index.…”

Section: Large Time Limitmentioning

confidence: 99%

“…We follow the formula of[8]. The formula obtained in[36] has a different form but recently[37] verified that the two formulas are equivalent.…”

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confidence: 99%

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Multipoint distribution of periodic TASEP

Baik

Liu

2019

J. Amer. Math. Soc.

158

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The height fluctuations of the models in the KPZ class are expected to converge to a universal process. The spatial process at equal time is known to converge to the Airy process or its variations. However, the temporal process, or more generally the two-dimensional space-time fluctuation field, is less well understood. We consider this question for the periodic TASEP (totally asymmetric simple exclusion process). For a particular initial condition, we evaluate the multi-time and multi-location distribution explicitly in terms of a multiple integral involving a Fredholm determinant. We then evaluate the large time limit in the so-called relaxation time scale.

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Section: Integrable Differential Equationsmentioning

confidence: 99%

Section: Large Time Limitmentioning

confidence: 99%

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Multipoint distribution of periodic TASEP

Baik

Liu

2019

J. Amer. Math. Soc.

158

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“…Recently it was shown that the height CDF at the (large time) KPZ fixed point for the full space problem is related to scale-invariant solutions of the Kadomtsev-Petviashvili (KP) equation [121]. A related observation was made for the periodic KPZ fixed point [122]. This connection to the KP equation extends to the generating function at arbitrary time for the KPZ equation in full space, for some particular initial conditions, droplet, half-Brownian [121] and Brownian [123].…”

Section: E Extended Kernel and The Kadomtsev-petviashvili Equationmentioning

confidence: 99%

Half-Space Stationary Kardar–Parisi–Zhang Equation

2020

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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.

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“…[92], by representing the polylogarithm as a sum of Hurwitz zeta functions (equation (57) in Ref. [92]), or as an infinite sum of square roots (equation (62) in Ref. [92]) with well-defined cuts.…”

Section: Large-activity Fluctuations Of the Edwards-wilkinson Regimementioning

confidence: 99%

Mapping current and activity fluctuations in exclusion processes: consequences and open questions

et al. 2021

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Considering the large deviations of activity and current in the Asymmetric Simple Exclusion Process (ASEP), we show that there exists a non-trivial correspondence between the joint scaled cumulant generating functions of activity and current of two ASEPs with different parameters. This mapping is obtained by applying a similarity transform on the deformed Markov matrix of the source model in order to obtain the deformed Markov matrix of the target model. We first derive this correspondence for periodic boundary conditions, and show in the diffusive scaling limit (corresponding to the Weakly Asymmetric Simple Exclusion Processes, or WASEP) how the mapping is expressed in the language of Macroscopic Fluctuation Theory (MFT). As an interesting specific case, we map the large deviations of current in the ASEP to the large deviations of activity in the SSEP, thereby uncovering a regime of Kardar--Parisi--Zhang in the distribution of activity in the SSEP. At large activity, particle configurations exhibit hyperuniformity [Jack et al., PRL 114, 060601 (2015)]. Using results from quantum spin chain theory, we characterize the hyperuniform regime by evaluating the small wavenumber asymptotic behavior of the structure factor at half-filling. Conversely, we formulate from the MFT results a conjecture for a correlation function in spin chains at any fixed total magnetization (in the thermodynamic limit). In addition, we generalize the mapping to the case of two open ASEPs with boundary reservoirs, and we apply it in the WASEP limit in the MFT formalism. This mapping also allows us to find a symmetry-breaking dynamical phase transition (DPT) in the WASEP conditioned by activity, from the prior knowledge of a DPT in the WASEP conditioned by the current.

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Riemann surfaces for KPZ with periodic boundaries

Cited by 23 publications

References 574 publications

Multipoint distribution of periodic TASEP

Multipoint distribution of periodic TASEP

Half-Space Stationary Kardar–Parisi–Zhang Equation

Mapping current and activity fluctuations in exclusion processes: consequences and open questions

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