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Register a new userLarge deviations of surface height in the $1+1$-dimensional Kardar-Parisi-Zhang equation: exact long-time results for $lambda H<0$
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We study atypically large fluctuations of height $H$ in the 1+1-dimensional Kardar-Parisi-Zhang (KPZ) equation at long times $t$, when starting from a droplet initial condition. We derive exact large deviation function of height for $lambda H<0$, where $lambda$ is the nonlinearity coefficient of the KPZ equation. This large deviation function describes a crossover from the Tracy-Widom distribution tail at small $|H|/t$, which scales as $|H|^3/t$, to a different tail at large $|H|/t$, which scales as $|H|^{5/2}/t^{1/2}$. The latter tail exists at all times $t>0$. It was previously obtained in the framework of the optimal fluctuation method. It was also obtained at short times from exact representation of the complete height statistics. The crossover between the two tails, at long times, occurs at $|H|sim t$ as previously conjectured. Our analytical findings are supported by numerical evaluations using exact representation of the complete height statistics.
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We use the optimal fluctuation method to evaluate the short-time probability distribution $mathcal{P}left(H,L,tright)$ of height at a single point, $H=hleft(x=0,tright)$, of the evolving Kardar-Parisi-Zhang (KPZ) interface $hleft(x,tright)$ on a ring of length $2L$. The process starts from a flat interface. At short times typical (small) height fluctuations are unaffected by the KPZ nonlinearity and belong to the Edwards-Wilkinson universality class. The nonlinearity, however, strongly affects the (asymmetric) tails of $mathcal{P}(H)$. At large $L/sqrt{t}$ the faster-decaying tail has a double structure: it is $L$-independent, $-lnmathcal{P}simleft|Hright|^{5/2}/t^{1/2}$, at intermediately large $|H|$, and $L$-dependent, $-lnmathcal{P}sim left|Hright|^{2}L/t$, at very large $|H|$. The transition between these two regimes is sharp and, in the large $L/sqrt{t}$ limit, behaves as a fractional-order phase transition. The transition point $H=H_{c}^{+}$ depends on $L/sqrt{t}$. At small $L/sqrt{t}$, the double structure of the faster tail disappears, and only the very large-$H$ tail, $-lnmathcal{P}sim left|Hright|^{2}L/t$, is observed. The slower-decaying tail does not show any $L$-dependence at large $L/sqrt{t}$, where it coincides with the slower tail of the GOE Tracy-Widom distribution. At small $L/sqrt{t}$ this tail also has a double structure. The transition between the two regimes occurs at a value of height $H=H_{c}^{-}$ which depends on $L/sqrt{t}$. At $L/sqrt{t} to 0$ the transition behaves as a mean-field-like second-order phase transition. At $|H|<|H_c^{-}|$ the slower tail behaves as $-lnmathcal{P}sim left|Hright|^{2}L/t$, whereas at $|H|>|H_c^{-}|$ it coincides with the slower tail of the GOE Tracy-Widom distribution.
We introduce a deterministic SO(3) invariant dynamics of classical spins on a discrete space-time lattice and prove its complete integrability by explicitly finding a related non-constant (baxterized) solution of the set-theoretic quantum Yang-Baxter equation over the 2-sphere. Equipping the algebraic structure with the corresponding Lax operator we derive an infinite sequence of conserved quantities with local densities. The dynamics depend on a single continuous spectral parameter and reduce to a (lattice) Landau-Lifshitz model in the limit of a small parameter which corresponds to the continuous time limit. Using quasi-exact numerical simulations of deterministic dynamics and Monte Carlo sampling of initial conditions corresponding to a maximum entropy equilibrium state we determine spin-spin spatio-temporal (dynamical) correlation functions with relative accuracy of three orders of magnitude. We demonstrate that in the equilibrium state with a vanishing total magnetization the correlation function precisely follow Kardar-Parisi-Zhang scaling hence the spin transport belongs to the universality class with dynamical exponent z=3/2, in accordance to recent related simulations in discrete and continuous time quantum Heisenberg spin 1/2 chains.
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. The short time behavior is investigated and compared with recent analytical predictions for the large-deviation forms of the probability of rare fluctuations, showing a spectacular agreement with the analytical expressions. The flat and stationary initial conditions are studied in the full space, together with the droplet initial condition in the half-space.
The joint probability distribution function (PDF) of the height and its gradients is derived for a zero tension $d+1$-dimensional Kardar-Parisi-Zhang (KPZ) equation. It is proved that the height`s PDF of zero tension KPZ equation shows lack of positivity after a finite time $t_{c}$. The properties of zero tension KPZ equation and its differences with the case that it possess an infinitesimal surface tension is discussed. Also potential relation between the time scale $t_{c}$ and the singularity time scale $t_{c, u to 0}$ of the KPZ equation with an infinitesimal surface tension is investigated.
We solve the large deviations of the Kardar-Parisi-Zhang (KPZ) equation in one dimension at short time by introducing an approach which combines field theoretical, probabilistic and integrable techniques. We expand the program of the weak noise theory, which maps the large deviations onto a non-linear hydrodynamic problem, and unveil its complete solvability through a connection to the integrability of the Zakharov-Shabat system. Exact solutions, depending on the initial condition of the KPZ equation, are obtained using the inverse scattering method and a Fredholm determinant framework recently developed. These results, explicit in the case of the droplet geometry, open the path to obtain the complete large deviations for general initial conditions.
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