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Many examples of natural systems can be described by random Gaussian surfaces. Much can be learned by analyzing the Fourier expansion of the surfaces, from which it is possible to determine the corresponding Hurst exponent and consequently establish the presence of scale invariance. We show that this symmetry is not affected by the distribution of the modulus of the Fourier coefficients. Furthermore, we investigate the role of the Fourier phases of random surfaces. In particular, we show how the surface is affected by a non-uniform distribution of phases.
Using the replica method, we develop an analytical approach to compute the characteristic function for the probability $mathcal{P}_N(K,lambda)$ that a large $N times N$ adjacency matrix of sparse random graphs has $K$ eigenvalues below a threshold $l
This review is devoted to the detailed consideration of the universal statistical properties of one-dimensional directed polymers in a random potential. In terms of the replica Bethe ansatz technique we derive several exact results for different type
We use Monte Carlo simulations to study the finite temperature behavior of vortices in the XY- model for tangent vector order on curved backgrounds. Contrary to naive expectations, we show that the underlying geometry does not affect the proliferatio
A well known connection between first-passage probability of random walk and distribution of electrical potential described by Laplace equation is studied. We simulate random walk in the plane numerically as a discrete time process with fixed step le
Statistical mechanics of 1D multivalent Coulomb gas may be mapped onto non-Hermitian quantum mechanics. We use this example to develop instanton calculus on Riemann surfaces. Borrowing from the formalism developed in the context of Seiberg-Witten dua