KPZ equation with a small noise, deep upper tail and limit shape

Lamarre, Pierre Yves Gaudreau; Lin, Yier; Tsai, Li-Cheng

doi:10.1007/s00440-022-01185-2

Probab. Theory Relat. Fields

2023

DOI: 10.1007/s00440-022-01185-2

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KPZ equation with a small noise, deep upper tail and limit shape

Pierre Yves Gaudreau Lamarre

Yier Lin

Li-Cheng Tsai

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Cited by 3 publications

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“…For the Freidlin-Wentzell/weak-noise LDP, behaviors of the one-point rate function and the corresponding most probable shapes for various initial conditions and boundary conditions have been predicted [KK07, KK08, KK09, MKV16, KMS16, MS17, MV18, SM18, SMS18, ALM19, SMV19], some of which have been proven [LT21, LT22, GLLT23]; an intriguing symmetry breaking and second-order phase transition has been discovered in [JKM16,SKM18] via numerical means and analytically derived in [KLD17,KLD22]; and a connection to integrable PDEs is established and explored in [Kra20, KLD21, KLD22, Tsa22b, KLD23]. The one-point upper-tail of the KPZ equation and the SHE have been studied in [Che15, CG20a, DT21, DG21, Lin21] for large time or all time larger than any given positive threshold, and in [GLLT23] for short time. For the nonlinear generalizations of the SHE, finite-time upper tails have been obtained in [CJK13, CD15, KKX17].…”

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Short Time Large Deviations of the KPZ Equation

Lin

Tsai

2021

Commun. Math. Phys.

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A. We consider the n-point, fixed-time large deviations of the KPZ equation with the narrow wedge initial condition. The scope consists of concave-configured, upper-tail deviations and a wide range of scaling regimes that allows time to be short, unit-order, and long. We prove the n-point large deviation principle and characterize, with proof, the corresponding spacetime limit shape. Our proof is based on the results -from the companion paper [Tsa23] -on moments of the stochastic heat equation and utilizes ideas coming from a tree decomposition. Behind our proof lies the phenomenon where the major contribution of the noise concentrates around certain corridors in spacetime, and we explicitly describe the corridors.

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Short Time Large Deviations of the KPZ Equation

Lin

Tsai

2021

Commun. Math. Phys.

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Short-time large deviations of the spatially averaged height of a Kardar–Parisi–Zhang interface on a ring

Schorlepp,

Sasorov,

Meerson

2023

J. Stat. Mech.

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Using the optimal fluctuation method, we evaluate the short-time probability distribution P ( H ˉ , L , t = T ) of the spatially averaged height H ˉ = ( 1 / L ) ∫ 0 L h ( x , t = T ) d x of a one-dimensional interface h ( x , t ) governed by the Kardar–Parisi–Zhang equation ∂ t h = ν ∂ x 2 h + λ 2 ∂ x h 2 + D ξ x , t on a ring of length L. The process starts from a flat interface, h ( x , t = 0 ) = 0 . Both at λ H ˉ < 0 and at sufficiently small positive λ H ˉ the optimal (that is, the least-action) path h ( x , t ) of the interface, conditioned on H ˉ , is uniform in space, and the distribution P ( H ˉ , L , T ) is Gaussian. However, at sufficiently large λ H ˉ > 0 the spatially uniform solution becomes sub-optimal and gives way to non-uniform optimal paths. We study these, and the resulting non-Gaussian distribution P ( H ˉ , L , T ) , analytically and numerically. The loss of optimality of the uniform solution occurs via a dynamical phase transition of either first or second order, depending on the rescaled system size ℓ = L / ν T , at a critical value H ˉ = H ˉ c ( ℓ ) . At large but finite ℓ the transition is of first order. Remarkably, it becomes an ‘accidental’ second-order transition in the limit of ℓ → ∞ , where a large-deviation behavior − ln P ( H ¯ , L , T ) ≃ ( L / T ) f ( H ¯ ) (in the units λ = ν = D = 1 ) is observed. At small ℓ the transition is of second order, while at ℓ = O ( 1 ) transitions of both types occur.

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Integrability in the weak noise theory

Tsai

2023

Trans. Amer. Math. Soc.

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We consider the variational problem associated with the Freidlin–Wentzell Large Deviation Principle (LDP) for the Stochastic Heat Equation (SHE). For a general class of initial-terminal conditions, we show that a minimizer of this variational problem exists, and any minimizer solves a system of imaginary-time Nonlinear Schrödinger equations. This system is integrable. Utilizing the integrability, we prove that the formulas from the physics work (see Alexandre Krajenbrink and Pierre Le Doussal [Phys. Rev. Lett. 127 (2021), p. 8]) hold for every minimizer of the variational problem. As an application, we consider the Freidlin–Wentzell LDP for the SHE with the delta initial condition. Under a technical assumption on the poles of the reflection coefficients, we prove the explicit expression for the one-point rate function that was predicted in the physics works (see Pierre Le Doussal, Satya N. Majumdar, Alberto Rosso, and Grégory Schehr [Phys. Rev. Lett. 117 (2016), p. 070403]; Alexandre Krajenbrink and Pierre Le Doussal [Phys. Rev. Lett. 127 (2021), p. 8]). Under the same assumption, we give detailed pointwise estimates of the most probable shape in the upper-tail limit.

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KPZ equation with a small noise, deep upper tail and limit shape

Cited by 3 publications

References 36 publications

Short Time Large Deviations of the KPZ Equation

Short Time Large Deviations of the KPZ Equation

Short-time large deviations of the spatially averaged height of a Kardar–Parisi–Zhang interface on a ring

Integrability in the weak noise theory

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