A point source on a plane constantly emits particles which rapidly diffuse and then stick to a growing cluster. The growth probability of a cluster is presented as a sum over all possible scenarios leading to the same final shape. The classical point for the action, defined as a minus logarithm of the growth probability, describes the most probable scenario and reproduces the Laplacian growth equation, which embraces numerous fundamental free boundary dynamics in non-equilibrium physics. For non-classical scenarios we introduce virtual point sources, in which presence the action becomes the Kullback-Leibler entropy. Strikingly, this entropy is shown to be the sum of electrostatic energies of layers grown per elementary time unit. Hence the growth probability of the presented non-equilibrium process obeys the Gibbs-Boltzmann statistics, which, as a rule, is not applied out from equilibrium. Each layer's probability is expressed as a product of simple factors in an auxiliary complex plane after a properly chosen conformal map. The action at this plane is a sum of Robin functions, which solve the Liouville equation. At the end we establish connections of our theory with the tau-function of the integrable Toda hierarchy and with the Liouville theory for non-critical quantum strings. The goal of this work is to unify two fundamental highly non-equilibrium processes, Laplacian growth (LG) [1][2][3][4][5], which is deterministic interface dynamics, and diffusion-limited aggregation (DLA) [6] -a discrete universal stochastic fractal growth. These remarkable processes have a lot in common and were suspected to be deeply related [7][8][9][10][11].Laplacian growth raised enormous interest in physics because of (i) its impressively wide applicability ranging from solidification and oil recovery to biological growth [1], (ii) remarkable universal asymptotic shapes, it exhibits [1,[12][13][14][15][16], and (iii) discoveries of deep intriguing connections of LG to quantum gravity [2] and the quantum Hall effect [17]. In mathematics the Laplacian growth appears so exciting because it possesses beautiful and powerful properties, unusual for most of nonlinear PDEs, such as infinitely many conservation laws [18] and closed form exact solutions [16,[19][20][21][22]. A new splash of intense activity in LG (see [3] for a review) was provoked by the work [2], where strong connections of LG with major integrable hierarchies and the theory of random matrices were established.Mathematical formulation of LG is (deceptively) simple: a droplet of air, D + (t), where t is time, is surrounded by a viscous fluid, D − (t) = C/D + (t), called D(t) for simplicity. Both liquids are sandwiched between two parallel close plates. Fluid velocity in D(t) obeys the Darcy law, v = −∇p (in scaled units), where p(z,z) is pressure and z = x + iy is a complex coordinate on the plane. Because of incompressibility, ∇·v = 0, then ∇ 2 p = 0 in D, except points with sources, which provide growth. Also, p = 0 at the interface, Γ(t) = ∂D(t), between two fluids, if...
We propose a free field representation for the form factors of descendant operators in the Bullough-Dodd model. This construction is a particular modification of Lukyanov's technique for solving the form factors axioms. We prove that the number of proposed solutions in each level subspace of the chiral sectors coincide with the number of the corresponding descendant operators in the Lagrangian formalism. We check that these form factors possess the cluster factorization property. Besides, we propose an alternative free field representation which allows us to study analytic properties of the form factors effectively. In particular, we prove that the form factors satisfy non trivial identities known as the "reflection relations". We show the existence of the reflection invariant basis in the level subspaces for a generic values of the parameters.
We develop statistical mechanics for stochastic growth processes and apply it to Laplacian growth by using its remarkable connection with a random matrix theory. The Laplacian growth equation is obtained from the variation principle and describes adiabatic (quasistatic) thermodynamic processes in the two-dimensional Dyson gas. By using Einstein's theory of thermodynamic fluctuations we consider transitional probabilities between thermodynamic states, which are in a one-to-one correspondence with simply connected domains occupied by gas. Transitions between these domains are described by the stochastic Laplacian growth equation, while the transitional probabilities coincide with a free-particle propagator on an infinite-dimensional complex manifold with a Kähler metric.
We generalize the diffusion-limited aggregation by issuing many randomly-walking particles, which stick to a cluster at the discrete time unit providing its growth. Using simple combinatorial arguments we determine probabilities of different growth scenarios and prove that the most probable evolution is governed by the deterministic Laplacian growth equation. A potential-theoretical analysis of the growth probabilities reveals connections with the tau-function of the integrable dispersionless limit of the two-dimensional Toda hierarchy, normal matrix ensembles, and the two-dimensional Dyson gas confined in a non-uniform magnetic field. We introduce the time-dependent Hamiltonian, which generates transitions between different classes of equivalence of closed curves, and prove the Hamiltonian structure of the interface dynamics. Finally, we propose a relation between probabilities of growth scenarios and the semi-classical limit of certain correlation functions of "light" exponential operators in the Liouville conformal field theory on a pseudosphere.
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