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Pierre Le Doussal - One of the best experts on this subject based on the ideXlab platform.

  • high precision simulation of the Height Distribution for the kpz equation
    EPL, 2018
    Co-Authors: Satya N Majumdar, Alexander K Hartmann, Pierre Le Doussal, Alberto Rosso, Gregory Schehr
    Abstract:

    The one-point Distribution of the Height for the continuum Kardar-Parisi-Zhang (KPZ) equation is determined numerically using the mapping to the directed polymer in a random potential at high temperature. Using an importance sampling approach, the Distribution is obtained over a large range of values, down to a probability density as small as 10^{-1000} in the tails. Both short and long times are investigated and compared with recent analytical predictions for the large-deviation forms of the probability of rare fluctuations. At short times the agreement with the analytical expression is spectacular. We observe that the far left and right tails, with exponents 5/2 and 3/2 respectively, are preserved until large time. We present some evidence for the predicted non-trivial crossover in the left tail from the 5/2 tail exponent to the cubic tail of Tracy-Widom, although the details of the full scaling form remains beyond reach.

  • exact short time Height Distribution in the one dimensional kardar parisi zhang equation with brownian initial condition
    Physical Review E, 2017
    Co-Authors: Alexandre Krajenbrink, Pierre Le Doussal
    Abstract:

    The early-time regime of the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimension, starting from a Brownian initial condition with a drift $w$, is studied using the exact Fredholm determinant representation. For large drift we recover the exact results for the droplet initial condition, whereas a vanishingly small drift describes the stationary KPZ case, recently studied by weak noise theory (WNT). We show that for short time $t$, the probability Distribution $P(H,t)$ of the Height $H$ at a given point takes the large deviation form $P(H,t)\ensuremath{\sim}exp\left[\ensuremath{-}\mathrm{\ensuremath{\Phi}}(H)/\sqrt{t}\right]$. We obtain the exact expressions for the rate function $\mathrm{\ensuremath{\Phi}}(H)$ for $Hl{H}_{c2}$. Our exact expression for ${H}_{c2}$ numerically coincides with the value at which WNT was found to exhibit a spontaneous reflection symmetry breaking. We propose two continuations for $Hg{H}_{c2}$, which apparently correspond to the symmetric and asymmetric WNT solutions. The rate function $\mathrm{\ensuremath{\Phi}}(H)$ is Gaussian in the center, while it has asymmetric tails, ${|H|}^{5/2}$ on the negative $H$ side and ${H}^{3/2}$ on the positive $H$ side.

  • tail of the two time Height Distribution for kpz growth in one dimension
    Journal of Statistical Mechanics: Theory and Experiment, 2017
    Co-Authors: Jacopo De Nardis, Pierre Le Doussal
    Abstract:

    Obtaining the exact multi-time correlations for one-dimensional growth models described by the Kardar-Parisi-Zhang (KPZ) universality class is presently an outstanding open problem. Here, we study the joint probability Distribution function (JPDF) of the Height of the KPZ equation with droplet initial conditions, at two different times $t_1Height at the earlier time $t_1$) is large and positive. Our formula interpolates between two limits where the JPDF decouples: (i) for $t_2/t_1 \to +\infty$ into a product of two GUE Tracy-Widom (TW) Distributions, and (ii) for $t_2/t_1 \to 1^+$ into a product of a GUE-TW Distribution and a Baik-Rains Distribution (associated to stationary KPZ evolution). The lowest cumulants of the Height at time $t_2$, conditioned on the one at time $t_1$, are expressed analytically as expansions around these limits, and computed numerically for arbitrary $t_2/t_1$. Moreover we compute the connected two-time correlation, conditioned to a large enough value at $t_1$, providing a quantitative prediction for the so-called persistence of correlations (or ergodicity breaking) in the time evolution from the droplet initial condition. Our RBA results are then compared with arguments based on Airy processes, with satisfactory agreement. These predictions are universal for all models in the KPZ class and should be testable in experiments and numerical simulations.

  • exact short time Height Distribution in the one dimensional kardar parisi zhang equation and edge fermions at high temperature
    Physical Review Letters, 2016
    Co-Authors: Pierre Le Doussal, Satya N Majumdar, Alberto Rosso, Gregory Schehr
    Abstract:

    : We consider the early time regime of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions in curved (or droplet) geometry. We show that for short time t, the probability Distribution P(H,t) of the Height H at a given point x takes the scaling form P(H,t)∼exp[-Φ_{drop}(H)/sqrt[t]] where the rate function Φ_{drop}(H) is computed exactly for all H. While it is Gaussian in the center, i.e., for small H, the probability Distribution function has highly asymmetric non-Gaussian tails that we characterize in detail. This function Φ_{drop}(H) is surprisingly reminiscent of the large deviation function describing the stationary fluctuations of finite-size models belonging to the KPZ universality class. Thanks to a recently discovered connection between the KPZ equation and free fermions, our results have interesting implications for the fluctuations of the rightmost fermion in a harmonic trap at high temperature and the full counting statistics at the edge.

P V Sasorov - One of the best experts on this subject based on the ideXlab platform.

  • short time Height Distribution in the one dimensional kardar parisi zhang equation starting from a parabola
    Physical Review E, 2016
    Co-Authors: Alex Kamenev, Baruch Meerson, P V Sasorov
    Abstract:

    : We study the probability Distribution P(H,t,L) of the surface Height h(x=0,t)=H in the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimension when starting from a parabolic interface, h(x,t=0)=x^{2}/L. The limits of L→∞ and L→0 have been recently solved exactly for any t>0. Here we address the early-time behavior of P(H,t,L) for general L. We employ the weak-noise theory-a variant of WKB approximation-which yields the optimal history of the interface, conditioned on reaching the given Height H at the origin at time t. We find that at small HP(H,t,L) is Gaussian, but its tails are non-Gaussian and highly asymmetric. In the leading order and in a proper moving frame, the tails behave as -lnP=f_{+}|H|^{5/2}/t^{1/2} and f_{-}|H|^{3/2}/t^{1/2}. The factor f_{+}(L,t) monotonically increases as a function of L, interpolating between time-independent values at L=0 and L=∞ that were previously known. The factor f_{-} is independent of L and t, signaling universality of this tail for a whole class of deterministic initial conditions.

  • short time Height Distribution in the one dimensional kardar parisi zhang equation starting from a parabola
    Physical Review E, 2016
    Co-Authors: Alex Kamenev, Baruch Meerson, P V Sasorov
    Abstract:

    We study the probability Distribution $\mathcal{P}(H,t,L)$ of the surface Height $h(x=0,t)=H$ in the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimension when starting from a parabolic interface, $h(x,t=0)={x}^{2}/L$. The limits of $L\ensuremath{\rightarrow}\ensuremath{\infty}$ and $L\ensuremath{\rightarrow}0$ have been recently solved exactly for any $tg0$. Here we address the early-time behavior of $\mathcal{P}(H,t,L)$ for general $L$. We employ the weak-noise theory---a variant of WKB approximation---which yields the optimal history of the interface, conditioned on reaching the given Height $H$ at the origin at time $t$. We find that at small $H\phantom{\rule{0.16em}{0ex}}\mathcal{P}(H,t,L)$ is Gaussian, but its tails are non-Gaussian and highly asymmetric. In the leading order and in a proper moving frame, the tails behave as $\ensuremath{-}ln\mathcal{P}={f}_{+}{|H|}^{5/2}/{t}^{1/2}$ and ${f}_{\ensuremath{-}}{|H|}^{3/2}/{t}^{1/2}$. The factor ${f}_{+}(L,t)$ monotonically increases as a function of $L$, interpolating between time-independent values at $L=0$ and $L=\ensuremath{\infty}$ that were previously known. The factor ${f}_{\ensuremath{-}}$ is independent of $L$ and $t$, signaling universality of this tail for a whole class of deterministic initial conditions.

Gregory Schehr - One of the best experts on this subject based on the ideXlab platform.

  • high precision simulation of the Height Distribution for the kpz equation
    EPL, 2018
    Co-Authors: Satya N Majumdar, Alexander K Hartmann, Pierre Le Doussal, Alberto Rosso, Gregory Schehr
    Abstract:

    The one-point Distribution of the Height for the continuum Kardar-Parisi-Zhang (KPZ) equation is determined numerically using the mapping to the directed polymer in a random potential at high temperature. Using an importance sampling approach, the Distribution is obtained over a large range of values, down to a probability density as small as 10^{-1000} in the tails. Both short and long times are investigated and compared with recent analytical predictions for the large-deviation forms of the probability of rare fluctuations. At short times the agreement with the analytical expression is spectacular. We observe that the far left and right tails, with exponents 5/2 and 3/2 respectively, are preserved until large time. We present some evidence for the predicted non-trivial crossover in the left tail from the 5/2 tail exponent to the cubic tail of Tracy-Widom, although the details of the full scaling form remains beyond reach.

  • exact short time Height Distribution in the one dimensional kardar parisi zhang equation and edge fermions at high temperature
    Physical Review Letters, 2016
    Co-Authors: Pierre Le Doussal, Satya N Majumdar, Alberto Rosso, Gregory Schehr
    Abstract:

    : We consider the early time regime of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions in curved (or droplet) geometry. We show that for short time t, the probability Distribution P(H,t) of the Height H at a given point x takes the scaling form P(H,t)∼exp[-Φ_{drop}(H)/sqrt[t]] where the rate function Φ_{drop}(H) is computed exactly for all H. While it is Gaussian in the center, i.e., for small H, the probability Distribution function has highly asymmetric non-Gaussian tails that we characterize in detail. This function Φ_{drop}(H) is surprisingly reminiscent of the large deviation function describing the stationary fluctuations of finite-size models belonging to the KPZ universality class. Thanks to a recently discovered connection between the KPZ equation and free fermions, our results have interesting implications for the fluctuations of the rightmost fermion in a harmonic trap at high temperature and the full counting statistics at the edge.

Baruch Meerson - One of the best experts on this subject based on the ideXlab platform.

  • Height Distribution tails in the kardar parisi zhang equation with brownian initial conditions
    Journal of Statistical Mechanics: Theory and Experiment, 2017
    Co-Authors: Baruch Meerson, Johannes Schmidt
    Abstract:

    For stationary interface growth, governed by the Kardar-Parisi-Zhang (KPZ) equation in 1 + 1 dimensions, typical fluctuations of the interface Height at long times are described by the Baik-Rains Distribution. Recently Chhita et al. [1] used the totally asymmetric simple exclusion process (TASEP) to study the Height fluctuations in systems of the KPZ universality class for Brownian interfaces with arbitrary diffusion constant. They showed that there is a one-parameter family of long-time Distributions, parametrized by the diffusion constant of the initial random Height profile. They also computed these Distributions numerically by using Monte Carlo (MC) simulations. Here we address this problem analytically and focus on the Distribution tails at short times. We determine the (stretched exponential) tails of the Height Distribution by applying the Optimal Fluctuation Method (OFM) to the KPZ equation. We argue that, by analogy with other initial conditions, the "slow" tail holds at arbitrary times and therefore provides a proper asymptotic to the family of long-time Distributions studied in Ref. [1]. We verify this hypothesis by performing large-scale MC simulations of a TASEP with a parallel-update rule. The "fast" tail, predicted by the OFM, is also expected to hold at arbitrary times, at sufficiently large Heights.

  • short time Height Distribution in the one dimensional kardar parisi zhang equation starting from a parabola
    Physical Review E, 2016
    Co-Authors: Alex Kamenev, Baruch Meerson, P V Sasorov
    Abstract:

    : We study the probability Distribution P(H,t,L) of the surface Height h(x=0,t)=H in the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimension when starting from a parabolic interface, h(x,t=0)=x^{2}/L. The limits of L→∞ and L→0 have been recently solved exactly for any t>0. Here we address the early-time behavior of P(H,t,L) for general L. We employ the weak-noise theory-a variant of WKB approximation-which yields the optimal history of the interface, conditioned on reaching the given Height H at the origin at time t. We find that at small HP(H,t,L) is Gaussian, but its tails are non-Gaussian and highly asymmetric. In the leading order and in a proper moving frame, the tails behave as -lnP=f_{+}|H|^{5/2}/t^{1/2} and f_{-}|H|^{3/2}/t^{1/2}. The factor f_{+}(L,t) monotonically increases as a function of L, interpolating between time-independent values at L=0 and L=∞ that were previously known. The factor f_{-} is independent of L and t, signaling universality of this tail for a whole class of deterministic initial conditions.

  • short time Height Distribution in the one dimensional kardar parisi zhang equation starting from a parabola
    Physical Review E, 2016
    Co-Authors: Alex Kamenev, Baruch Meerson, P V Sasorov
    Abstract:

    We study the probability Distribution $\mathcal{P}(H,t,L)$ of the surface Height $h(x=0,t)=H$ in the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimension when starting from a parabolic interface, $h(x,t=0)={x}^{2}/L$. The limits of $L\ensuremath{\rightarrow}\ensuremath{\infty}$ and $L\ensuremath{\rightarrow}0$ have been recently solved exactly for any $tg0$. Here we address the early-time behavior of $\mathcal{P}(H,t,L)$ for general $L$. We employ the weak-noise theory---a variant of WKB approximation---which yields the optimal history of the interface, conditioned on reaching the given Height $H$ at the origin at time $t$. We find that at small $H\phantom{\rule{0.16em}{0ex}}\mathcal{P}(H,t,L)$ is Gaussian, but its tails are non-Gaussian and highly asymmetric. In the leading order and in a proper moving frame, the tails behave as $\ensuremath{-}ln\mathcal{P}={f}_{+}{|H|}^{5/2}/{t}^{1/2}$ and ${f}_{\ensuremath{-}}{|H|}^{3/2}/{t}^{1/2}$. The factor ${f}_{+}(L,t)$ monotonically increases as a function of $L$, interpolating between time-independent values at $L=0$ and $L=\ensuremath{\infty}$ that were previously known. The factor ${f}_{\ensuremath{-}}$ is independent of $L$ and $t$, signaling universality of this tail for a whole class of deterministic initial conditions.

Alex Kamenev - One of the best experts on this subject based on the ideXlab platform.

  • short time Height Distribution in the one dimensional kardar parisi zhang equation starting from a parabola
    Physical Review E, 2016
    Co-Authors: Alex Kamenev, Baruch Meerson, P V Sasorov
    Abstract:

    : We study the probability Distribution P(H,t,L) of the surface Height h(x=0,t)=H in the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimension when starting from a parabolic interface, h(x,t=0)=x^{2}/L. The limits of L→∞ and L→0 have been recently solved exactly for any t>0. Here we address the early-time behavior of P(H,t,L) for general L. We employ the weak-noise theory-a variant of WKB approximation-which yields the optimal history of the interface, conditioned on reaching the given Height H at the origin at time t. We find that at small HP(H,t,L) is Gaussian, but its tails are non-Gaussian and highly asymmetric. In the leading order and in a proper moving frame, the tails behave as -lnP=f_{+}|H|^{5/2}/t^{1/2} and f_{-}|H|^{3/2}/t^{1/2}. The factor f_{+}(L,t) monotonically increases as a function of L, interpolating between time-independent values at L=0 and L=∞ that were previously known. The factor f_{-} is independent of L and t, signaling universality of this tail for a whole class of deterministic initial conditions.

  • short time Height Distribution in the one dimensional kardar parisi zhang equation starting from a parabola
    Physical Review E, 2016
    Co-Authors: Alex Kamenev, Baruch Meerson, P V Sasorov
    Abstract:

    We study the probability Distribution $\mathcal{P}(H,t,L)$ of the surface Height $h(x=0,t)=H$ in the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimension when starting from a parabolic interface, $h(x,t=0)={x}^{2}/L$. The limits of $L\ensuremath{\rightarrow}\ensuremath{\infty}$ and $L\ensuremath{\rightarrow}0$ have been recently solved exactly for any $tg0$. Here we address the early-time behavior of $\mathcal{P}(H,t,L)$ for general $L$. We employ the weak-noise theory---a variant of WKB approximation---which yields the optimal history of the interface, conditioned on reaching the given Height $H$ at the origin at time $t$. We find that at small $H\phantom{\rule{0.16em}{0ex}}\mathcal{P}(H,t,L)$ is Gaussian, but its tails are non-Gaussian and highly asymmetric. In the leading order and in a proper moving frame, the tails behave as $\ensuremath{-}ln\mathcal{P}={f}_{+}{|H|}^{5/2}/{t}^{1/2}$ and ${f}_{\ensuremath{-}}{|H|}^{3/2}/{t}^{1/2}$. The factor ${f}_{+}(L,t)$ monotonically increases as a function of $L$, interpolating between time-independent values at $L=0$ and $L=\ensuremath{\infty}$ that were previously known. The factor ${f}_{\ensuremath{-}}$ is independent of $L$ and $t$, signaling universality of this tail for a whole class of deterministic initial conditions.