We investigate growing interfaces of topological-defect turbulence in the electroconvection of nematic liquid crystals. The interfaces exhibit self-affine roughening characterized by both spatial and temporal scaling laws of the Kardar-Parisi-Zhang theory in 1+1 dimensions. Moreover, we reveal that the distribution and the two-point correlation of the interface fluctuations are universal ones governed by the largest eigenvalue of random matrices. This provides quantitative experimental evidence of the universality prescribing detailed information of scale-invariant fluctuations.

Stochastic motion of a point – known as Brownian motion – has many successful applications in science, thanks to its scale invariance and consequent universal features such as Gaussian fluctuations. In contrast, the stochastic motion of a line, though it is also scale-invariant and arises in nature as various types of interface growth, is far less understood. The two major missing ingredients are: an experiment that allows a quantitative comparison with theory and an analytic solution of the Kardar-Parisi-Zhang (KPZ) equation, a prototypical equation for describing growing interfaces. Here we solve both problems, showing unprecedented universality beyond the scaling laws. We investigate growing interfaces of liquid-crystal turbulence and find not only universal scaling, but universal distributions of interface positions. They obey the largest-eigenvalue distributions of random matrices and depend on whether the interface is curved or flat, albeit universal in each case. Our exact solution of the KPZ equation provides theoretical explanations.

We provide a comprehensive report on scale-invariant fluctuations of growing interfaces in liquid-crystal turbulence, for which we recently found evidence that they belong to the Kardar-Parisi-Zhang (KPZ) universality class for 1 + 1 dimensions [Takeuchi and Sano in Phys. Rev. Lett. 104:230601, 2010; Takeuchi et al. in Sci. Rep. 1:34, 2011]. Here we investigate both circular and flat interfaces and report their statistics in detail. First we demonstrate that their fluctuations show not only the KPZ scaling exponents but beyond: they asymptotically share even the precise forms of the distribution function and the spatial correlation function in common with solvable models of the KPZ class, demonstrating also an intimate relation to random matrix theory. We then determine other statistical properties for which no exact theoretical predictions were made, in particular the temporal correlation function and the persistence probabilities. Experimental results on finite-time effects and extreme-value statistics are also presented. Throughout the paper, emphasis is put on how the universal statistical properties depend on the global geometry of the interfaces, i.e., whether the interfaces are circular or flat. We thereby corroborate the powerful yet geometry-dependent universality of the KPZ class, which governs growing interfaces driven out of equilibrium.

The stochastic partial differential equation proposed nearly three decades ago by Kardar, Parisi and Zhang (KPZ) continues to inspire, intrigue and confound its many admirers. Here, we i) pay debts to heroic predecessors, ii) highlight additional, experimentally relevant aspects of the recently solved 1+1 KPZ problem, iii) use an expanding substrates formalism to gain access to the 3d radial KPZ equation and, lastly, iv) examining extremal paths on disordered hierarchical lattices, set our gaze upon the fate of d=∞ KPZ. Clearly, there remains ample unexplored territory within the realm of KPZ and, for the hearty, much work to be done, especially in higher dimensions, where numerical and renormalization group methods are providing a deeper understanding of this iconic equation.Keywords Nonequilibrium Growth · Extremal Paths · Universal Limit Distributions In a NutshellThe history of physics has been punctuated at seminal moments by the appearance of certain fundamental equations (and associated models) which have vigorously propelled the enterprise forward, serving as an explosively rich departure point, generating a myriad of alternative perspectives, creative insights, surprising connections and, given its sustained impregnability, often remain for many years a sacred object of fascination and obsession to its dedicated disciples. Recent, obvious suspects in this regard include the quantum mechanical Schrödinger equation (equally well, its flipside-Feynman's path integral formulation), or the wonderfully elusive Navier-Stokes equation governing fluid mechanics, by which may be gleaned the scaling secrets of turbulent flow, dynamically encoded in the whirling eddies drawn by da Vinci centuries ago. Within the domain of equilibrium statistical physics, the 2d Ising Model, with its tour-de-force algebraic solution by Onsager, followed by combinatoric, graphical, Grassmannian, Monte Carlo, as well as full-blown field-theoretic, scaling, and renormalization group treatments, represents an extraordinary legacy that continues unabated to this very day. Arguably, a non-equilibrium statistical mechanical analogue to Ising/Onsager is the iconic equation [1] proposed a generation ago by Kardar, Parisi, and Zhang (KPZ); it captures the statistical fluctuations of a kinetically-roughened scalar height field h(x, t):