{'fraction_non_alphanumeric': 0.0605171794564827, 'fraction_numerical': 0.014725248413743565, 'mean_word_length': 4.332588573252473, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Stochastic Laplacian growth

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 fluid normal velocity at the interface, thus where ∂ n is a normal derivative. Diffusion-limited aggregation is a process where equal particles are issued one by one from infinity and diffuse until they stick to a growing cluster [6]. Remarkably, all grown clusters are monofractals with the numerically obtained Hausdorff dimension D h = 1.71 ± 0.01, which appears to be robust and universal [25] (independent on geometrical details). Analytic derivation of this number remains a long-standing challenge in non-equilibrium physics despite numerous efforts [25]. Surprisingly, the same fractal dimension was observed in several Laplacian growth experiments [26,27], where the process is continuous and deterministic.
We have unifed LG and DLA as two opposite limits (classical and quantum respectively) of a stochastic Laplacian growth, where instead of one particle the source emits K ≥ 1 uncorrelated particles per time unit. The DLA, when K = 1, can be called a quantum limit of this process, as correlations between particles in this case are maximal. The next particle always "feels" a slight change of the interface, caused by a previously landed particle, while both would be totally uncorrelated if emitted simultaneously.
By using simple combinatorics we introduce below probability P of different growth scenarios and define the action as A = − log P, where is the particle area. Then we show that in the limit, K → ∞, the most probable motion of D(t) (the classical point of the action) is deterministic and obeys the Laplacian growth equation. Thus, K → ∞, is the classical limit of this theory.
It is valuable that the action for Laplacian growth comes so simply from growth probability. For it is known that to find a functional, which extremum gives equations for dissipative motion, is much harder, than for frictionless processes, whose Lagrangian or Hamiltonian structure is often straightforward [44]. So far the Laplacian growth equation was derived only as the approximation of viscous hydrodynamics.
Electrostatics and Gibbs-Boltzmann statistics. We found that the action for an arbitrary D(T ) is fully characterized by harmonic measures, µ n (A m ), for sources at A m and by their strengths, Q m . Then we derived that the action equals a time integral from the entropy, which is LHS of (2) below. Surprisingly, this purely probabilistic expression can be transformed to a sum of electrostatic potentials created at a m (the conformal image of A m ) by charges induced on the unit circle, kept at zero potential: (2) From (2) it follows that growth probabilities for these patterns obey the usual equilibrium Gibbs-Boltzmann distribution: (3) where K serves as temperature. This conclusion opens novel possibilities for analyzing non-equilibrium growth processes by tools of equilibrium statistical physics.
Structure of the paper is straightforward: after introducing harmonic measure and conformal map we derive the classical Laplacian growth (9) from elementary probability formulae and introduce virtual classical sources, which cause "non-classical" complex shapes. This helps to present the growth probability of non-classical shapes in terms of classical sources (14). Then we transform the entropy (14) to electrostatic energy (21) in the "conformal" w-plane and (26) in the physical z-plane. Finally, we reveal physical significance of growth probabilities (21) and (26), which are two main results of this work, and establish connections with modern mathematical physics.
The harmonic measure, µ D (ξ n , A), is important in what follows. For simplicity we will skip below the label D and will often refer to µ D (ξ n , A) just as µ n (A). Let's partition the boundary, Γ, into N 1 little fragments of the size |dξ n |, so n = 1, 2, . . . , N . Then the harmonic measure, µ n (A), for the n-th fragment between ξ n and ξ n + dξ n , with the source at A, is defined [30] as where ∂ n is a normal derivative, and G D (z, ζ) is the Green function of the domain D. By definition, G D (z, ζ) is a harmonic function in D, except at z = ζ, where G(z, ζ) diverges as log |z − ζ| [33], and also G D (z, ζ) = 0 at the boundary Γ. In electrostatics µ n (A) is a charge distribution induced at ξ n ∈ Γ by a unit charge at A to keep Γ equipotential. But in this work the harmonic measure is a probability for a Brownian particle, issued at A ∈ D(t), to land between ξ n and ξ n + dξ n at Γ. Conformal mapping. Harmonic nature of G D(t) (z, A) suggests the (time dependent) conformal mapping, z = f (w), from the exterior of the unit circle at the auxiliary complex w-plane to the domain D(t) at the physical zplane. Then the unit circle, w = e iφ , maps to Γ(t): ξ = f (e iφ ) ∈ Γ(t). Setting ∞ → ∞ and f (∞) > 0 makes this map unique. Each fragment |dξ| ∈ Γ is mapped from a little arc dφ lying between φ and φ+dφ at the unit circle, and the source A = f (1/ā), where a is a singularity of f (w), and |a| < 1 (see Fig. 1). Calculating µ n (A) is simple at w-plane, chosen instead of z-plane, since µ n (A) is conformally invariant. From the Green function outside the unit circle, G(w, 1/ā) = log(|1 −āw|/|w − a|), and (4) we obtain (5) Stochastic Laplacian growth, 1 unifying LG and DLA as two opposite limits, differs from DLA by K ≥ 1 (instead of one) of simultaneously issued independent particles of area from several sources. The particles are curvilinear quadrangles formed by equipotential and stream lines, generated by the probability field. A growing domain is initially a unit circle, so 1. The particles quickly diffuse until they stick to a growing cluster (forming on its surface an external layer of the area K ) per characteristic time, δt, which determines a timescale (a unit time) in the problem. Since they diffuse and stick at the interface much faster than the interface grows, δt can be treated as the small time interval. The K issued uncorrelated particles are to be distributed into N bins of the boundary with the probabilities, expressed by the harmonic measure (4), such that the particles stuck to the Since these particle sources become fluid sinks with rates Q m in the continuous (hydrodynamical) limit, the corresponding partial area increments are equal: A single layer, grown by K particles issued from M sources at A m , is defined by k = {k m1 , . . . , k mN } M m=1 , where k mn is a number of particles deposited from A m and landed to the n-th fragment. Its probability is given by the multinomial formula, P (k) = M m=1 K m ! N n=1 µ n (A m ) kmn /k mn !, which in the Stirling approximation, K m 1, takes a form of the Kullback-Leibler entropy [29] (the distance between two distributions): Derivation of the Laplacian growth equation, (eqs. (9) and (10) below). Variation of (7) with the constraints, K m = N n=1 k nm , shows that P is maximal when This maximum is exponentially sharp when → 0, so all fluctuations around k * mn are suppressed. Hence k * mn is a classical trajectory for this stochastic process. It describes the deterministic Laplacian growth with M sources, Q m at A m = f (1/ā m ). Indeed, because the normal displacement at ξ n is V (ξ n )δt = √ m k * mn (the particles are approximate squares, √ × √ ), it readily follows from (5), (6), (8), and the identity, which is the classical Laplacian growth with M sources. If M = 1, Q 1 (t) = Q, and A 1 = ∞, (9) takes a form which was intensely studied earlier [1-5, 31] and addressed by the stochastic LG in [32].
where P (k i+1 ) is a conditional probability for the layer k i+1 to grow over the domain D i ≡ D(iδt). Since P (k i+1 ) is independent of prehistory, t < iδt, the product in (11) is a Markovian chain, and the sum over all scenarios extends standard path-integration to scenarios in the space {Γ(t)} × [0, T ). Taking P (k i ) from (7), K m from (6), and transforming (11), we obtain, when δt → 0, that P(k) = exp{−A(k)/ }, where we defined the stochastic action A(k) as (12) For a single source at infinity, M = 1 and A 1 = ∞, the action (12) takes a form where we replaced k * 1n by its classical value (8).
Virtual sources. We see from (10) that in LG with a single source at infinity an initial circle, z = r 0 e iφ , stays as a circle, z = r(t)e iφ . It is the classical trajectory of the stochastic action (13). But in experiments with a single source far enough from Γ(t) to be treated as at infinity [12,26,27], complex irregular interfaces, caused by intrinsic instabilities of the process, are always observed.
Remarkably, any non-circular domain D(T ) bears "fingerprints" (singularities of the Schwarz function [45], S(T, z) for Γ(T ), lying in D(T )) left by sources operated at earlier times, t < T . These complex shapes, D(T ), correspond to non-classical trajectories of the stochastic action (13). Thus, all deviations of "non-classical" D(T ) from a growing circle we attribute to these virtual sources at A m , working in their classical regime [46]. This process is described by the classical equation (9), but with time-dependent virtual sources, Q m (except Q 1 at ∞), which cause observable deflections of Γ(t) from the classical path, prescribed by (10) [47]. Each virtual source A m contributes to the growth probability by the action (13) with k n given by (8). By summing up contributions of the independent virtual sources during δt we obtain the logarithm of the probability of the non-classical layer: where µ n,i (A m ) is referred to D i . This is the Kullback-Leibler entropy mentioned above.
Contribution of log(K/K mi ) in (14) Here N i is the number of partitions of K particles into M i groups (sources), K 1 , . . . , K Mi , at i-th time step. Replacing µ n,i (A m ) by − ∂ n G i (ξ, A m ) |dξ|/(2π) under the logarithm in (14) we rewrite (14) in the continuous limit, |dξ n | → 0, as A Dirichlet problem of recovery harmonic functions by their boundary values is of great help in calculating this integral. Notice that ∂ n G(ξ, ∞) = −|w (ξ)|, where w = w(z) is inverse to the conformal map, z = f (w), introduced earlier. Since log |w (z)| is harmonic in D, then log |w (ξ)| is the boundary value of this harmonic function, so the contribution of the numerator to the integral in (15) equals  W (z, A) as G(z, A) (17) Since near infinity ∂ z W − (ξ, A) = −1/ξ + O(ξ −2 ), then subtracting G(ξ, ∞) = 0 at Γ from (17) makes the expression, log |1 + (ξ − A) ∂ z W − (ξ, A)| − G(ξ, ∞), harmonic for ξ everywhere in D. Therefore the corresponding integral in (15), being a solution of the Dirichlet boundary problem, is the difference between Γ µ(ξ, A) log |ξ − A| = −G − (A, A) + G(A, ∞) from the denominator and (17). After adding (16) to this difference we finally obtain the remarkable identity, which has clear electrostatic interpretation shown below.
In w-plane the integral (18) can be further simplified. Namely, because of (5), the integral (18) is easily calculated to equal where A = f (1/ā), as said above. This is the Robin function [34], which is a potential at a created by charges induced by a unit charge at a on the unit circle, kept at zero potential. Thus, the entropy (14) was transformed to electrostatic energy, and with help of (19) the probability (15) for a single layer can be compactly rewritten in a form of the Gibbs-Boltzmann distribution, (implying that probability P i = C exp(−βE i ), where E i is the energy of i-th state and β is a positve constant): Then in the limit δt → 0 the scenario probability (11) becomes (after defining N = and A m = f (1/ā m ) provides time-dependence of a m .
In z-plane the integral (18) allows to recast (15) in another remarkable way: adding (15) over all i and assuming one source per one time unit (we set m = i for convenience) we obtain via (11) the logarithm of probability of the given scenario, (22) Contribution from the denominator in the RHS of (22) equals to the following neat expression, which depends on the final domain D(T ) only, but not on a particular way to arrive to it [35]: log |w T (ξ)|ξ dξ i .
(23) As to the numerator in (22), it is shown [35] to equal which is the energy of self-interacting charge induced on Γ i−1 with density µ(ξ, A i ). After transforming contour integrals over Γ i−1 into integrals over the layer, l i = D i /D i−1 , it becomes (see [35]) where A i (z) = |z| 2 /2−Re z 0 S(iδt, z ) dz is the so-called modified Schwarz potential [36]. Then the probability of the whole scenario (11) equals Strikingly, despite non-equilibrium nature of LG, in both w and z planes, the growth probabilities (21) and (26), appear in a form of the equilibrium Gibbs-Boltzmann distribution, since exponents in these formulae are electrostatic energies (up to multiplicative constants). Finally, interesting connections were found between the stochastic LG and modern mathematical physics.
Connection to the Liouville theory. Remarkably, the Robin function, G − (w, w) = − log(1 − |w| 2 ), obtained in (19), obeys the (integrable) Liouville equation, vital for the theory of non-critical strings [37], Thus, the probability (20) appears to be the classical limit of the multipoint correlation function of "light" exponential operators in the Liouville theory on the pseudosphere [38] (see [35] for details).
Connection to the tau-function for the Toda hierarchy. The main result (26) for a scenario probability can easily be rewritten through the following integrals which coincide with the tau-function for analytic curves [39] after replacing a layer l i by a simply connected finite domain D. The obtained expression (skipped here for want of space) connects growth probability to the dispersionless 2D integrable Toda hierarchy. Replacement of l i by D in (28) implies interaction of all layers constituting the domain. However, in our case the layers, l i , enter (26) additively, and so are independent. Hence the sources, participated in growth of different layers, are also mutually independent (but not commutative, contrary to the classical deterministic LG).
Summarizing, we emphasize the derivation of the Laplacian growth equation (9) from the action functional using elementary combinatorics, and unexpected relation between the entropy (14) and electrostatic energies on the w-plane (19) and the z-plane (24). As the result, the growth probabilities satisfy the Gibbs-Boltzmann statistics, suggesting applications of weakly non-equilibrium thermodynamics to this highly unstable and non-equilibrium process.
In conclusion, we state the expected impact of the results, obtained in this paper, -for LG and DLA: The presented theory promises to elucidate derivation of the DLA fractal spectrum and unexplained selection problems in LG.
-for non-equilibirum physics: It appears possible now to address highly non-equlibrium growth in the framework of linear non-equilibrium thermodynamics [43] and to reinterpret complex pattern formation as a selforganizing non-equilibrium process.
-for other branches of physics: Remarkably, the growth probability (26) links stochastic LG with a growth of an electronic droplet in a quantum Hall effect [17]. It allows us to reformulate our model in terms of normal random matrices [41], which underly 2D quantum gravity [42].
The next step is to go beyond the classical limit, which was the subject of this work, and to study quantum stochastic Laplacian growth, where the correlations between particles become important, i.e. when K is small.