We study multiplicative statistics for the eigenvalues of unitarily-invariant Hermitian random matrix models. consider one-cut regular polynomial potentials and a large class statistics. show that in limit several associated quantities converge to limits which are universal both potential family considered. In turn, such described by integro-differential Painlev\'e II equation, particular they ...

The logarithmic derivative of the marginal distributions randomly fluctuating interfaces in one dimension on a large scale evolve according to Kadomtsev-Petviashvili (KP) equation. This is derived algebraically from Fredholm determinant obtained [MQR17, arXiv:1701.00018] for KPZ fixed point as limit transition probabilities TASEP, special solvable model universality class. Tracy-Widom appear se...

This paper is concerned with the construction of atomic Gaussian multiplicative chaos and the KPZ formula in Liouville quantum gravity. On the first hand, we construct purely atomic random measures corresponding to values of the parameter γ2 beyond the transition phase (i.e. γ2 > 2d) and check the duality relation with sub-critical Gaussian multiplicative chaos. On the other hand, we give a sim...

We consider the half-line stochastic heat equation (SHE) with Robin boundary parameter $A = -\frac{1}{2}$. Under narrow wedge initial condition, we compute every positive (including non-integer) Lyapunov exponents of SHE. As a consequence, prove large deviation principle for upper tail KPZ under Neumann -\frac{1}{2}$ rate function $\Phi_+^{\text{hf}} (s) \frac{2}{3} s^{\frac{3}{2}}$. This confi...

The power spectrum of interface fluctuations in the (1 + 1)-dimensional Kardar-Parisi-Zhang (KPZ) universality class is studied both experimentally and numerically. The 1/f-type spectrum is found and characterized through a set of “critical exponents” for the power spectrum. The recently formulated “aging WienerKhinchin theorem” accounts for the observed exponents. Interestingly, the 1/f spectr...

A large class of physically important nonlinear and nonhomogeneous evolution problems, characterized by advection-like and diffusion-like processes, can be usefully studied by a time-differential form of Kolmogorov’s solution of the backward-time Fokker-Planck equation. The differential solution embodies an integral representation theorem by which any physical or mathematical entity satisfying ...

Motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions, like the Kardar-Parisi-Zhang (KPZ) equation, we reconsider the unique solvability of one-dimensional stochastic differential equations, the drift of which is a distribution, by means of rough paths theory. Existence and uniqueness are established in the wea...

We study two coupled spatially extended dynamical systems which exhibit space-time chaos. The transition to the synchronized state is treated as a nonequilibrium phase transition, where the average synchronization error is the order parameter. The transition in one-dimensional systems is found to be generically in the universality class of the Kardar-Parisi-Zhang equation with a growth-limiting...