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We show that the problem of a directed polymer on a tree with disorder can be reduced to the study of nonlinear equations of reaction-diffusion type. These equations admit traveling wave solutions that move at all possible speeds above a certain minimal speed. The speed of the wavefront is the free energy of the polymer problem and the minimal speed corresponds to a phase transition to a glassy phase similar to the spin-glass phase. Several properties of the polymer problem can be extracted from the correspondence with the traveling wave: probability distribution of the free energy, overlaps, etc.

We develop a scaling theory for KPZ growth in one dimension by a detailed study of the polynuclear growth (PNG) model. In particular, we identify three universal distributions for shape fluctuations and their dependence on the macroscopic shape. These distribution functions are computed using the partition function of Gaussian random matrices in a cosine potential.PACS numbers: 64.60. Ht, 68.35.Ct, 81.10.Aj Growth processes lead to a rich variety of macroscopic patterns and shapes [1]. As has been recognized for some time, growth may also give rise to intriguing statistical fluctuations comparable to thermal fluctuations at a critical point. One of the most prominent examples is the Kardar-Parisi-Zhang (KPZ) universality class [2]. In essence one models a stable phase which grows into an unstable phase through aggregation, as for example in Eden type models where perimeter sites of a given cluster are filled up randomly. In real materials, mere aggregation is often too simplistic an assumption and one would have to take other dynamical modes, such as surface diffusion, at the stable/unstable interface into account [3]. In our letter we remain within the KPZ class.From the beginning there has been evidence that in one spatial dimension KPZ growth processes are linked to exactly soluble models of two-dimensional statistical mechanics. Kardar [4] mapped growth to the directed polymer problem. The replica trick then yields the Bose gas with attractive δ-interaction which in one dimension can be solved through the Bethe ansatz [5]. In [6], considerably generalized in [7], for a particular discrete growth model the statistical weights for the local slopes were mapped onto the six vertex model. To solve the six vertex model one diagonalizes the transfer matrix, again, through the Bethe ansatz, which also allows for a study of finite size scaling [8]. Unfortunately none of these methods go beyond what corresponds to the free energy in the six vertex model and the associated dynamical scaling exponent β = 1/3.In this letter we point out that within the KPZ universality class the polynuclear growth (PNG) model plays a distinguished role: it maps onto random permutations, the height being the length of the longest increasing subsequence of such a permutation, and thereby onto Gaussian random matrices [9,10]. We use these mappings to obtain an analytic expression for certain scaling distributions, which then leads to an understanding of how the self-similar height fluctuations depend on the initial conditions and to a more refined scaling theory for KPZ growth.PNG is a simplified model for layer by layer growth [1]. One starts with a perfectly flat crystal in contact with its super-saturated vapor. Once in a while a supercritical nucleus is formed, which then spreads laterally by further attachment of particles at its perimeter sites. Such islands coalesce if they are in the same layer and further islands may be nucleated upon already existing ones. The PNG model ignores the lateral lattice structure and assumes that the...

With focus on anharmonic chains, we develop a nonlinear version of fluctuating hydrodynamics, in which the Euler currents are kept to second order in the deviations from equilibrium and dissipation plus noise are added. The required model-dependent parameters are written in such a way that they can be computed numerically within seconds, once the interaction potential, pressure, and temperature are given. In principle the theory is applicable to any one-dimensional system with local conservation laws. The resulting nonlinear stochastic field theory is handled in the one-loop approximation. Some of the large scale predictions can still be worked out analytically. For more details one has to rely on numerical simulations of the corresponding mode-coupling equations. In this way we arrive at detailed predictions for the equilibrium time correlations of the locally conserved fields of an anharmonic chain.

PrefaceBy intention, my project has two parts. The first one covers the classical electron theory. It is essentially self-contained and will be presented in the following chapters. 75 years after the discovery of quantum mechanics, to discuss only the classical version of the theory looks somewhat obsolete, in particular since many phenomena, like the stability of atoms, the existence of spectral lines and their life time, the binding of atoms, and many others, are described only by the quantized theory. Thus it is a necessity to discuss the quantized version of the classical models studied here. This is not quantum electrodynamics. It is the quantum theory of electrons, stable nuclei, and photons with no pair production allowed. Well said, but the quantum part turns out to be a difficult task. There is a lot of material with the mathematical physics side in flux and very active at present. Thus it remains to be seen whether the quantum part will be ever finished. In the meantime I invite the reader to comments, criticisms, and improvements on the classical part.In thank my collaborators, Sasha Komech and Markus Kunze, for their constant help and insistence. I am very grateful to Joel Lebowitz. He initiated my interest in tracer particle problems and I always wanted to apply these ideas to electrons coupled to the Maxwell field. I am indebted to F. Rohrlich for discussions and important hints on the literature. I acknowledge instructive discussions with A.

We report on the first exact solution of the Kardar-Parisi-Zhang (KPZ) equation in one dimension, with an initial condition which physically corresponds to the motion of a macroscopically curved height profile. The solution provides a determinantal formula for the probability distribution function of the height h(x,t) for all t>0. In particular, we show that for large t, on the scale t(1/3), the statistics is given by the Tracy-Widom distribution, known already from the Gaussian unitary ensemble of random matrix theory. Our solution confirms that the KPZ equation describes the interface motion in the regime of weak driving force. Within this regime the KPZ equation details how the long time asymptotics is approached.