Self-Organized Criticality and Pattern Emergence Through the Lens of Tropical Geometry

Self-Organized Criticality and Pattern Emergence through the lens of Tropical Geometry

N. Kalinin,1 A. Guzman-S´ aenz,´ 2 Y. Prieto, 3 M. Shkolnikov, 4 V. Kalinina,5 E. Lupercio 6∗

1Higher School of Economics, The Laboratory of Game Theory and Decision Making Saint-Petersburg, Russia 2Computational Genomics, IBM T.J. Watson Research Center, Yorktown Heights, NY, USA 3Institut de Mathematiques´ de Toulouse, Universite´ de Toulouse III Paul Sabatier Toulouse, France 4Institute of Science and Technology Austria Klosterneuburg, Austria 5Institute of Cytology, Russian Academy of Science Saint-Petersburg, Russia 6CINVESTAV, Department of Mathematics Campus Zacatenco, CDMX, Mexico ∗To whom correspondence should be addressed; e-mail: [email protected]

Tropical Geometry, an established ﬁeld in pure mathematics, is a place where String Theory, Mirror Symmetry, Computational Algebra, Auction Theory, arXiv:1806.09153v1 [nlin.AO] 24 Jun 2018 etc, meet and inﬂuence each other. In this paper, we report on our discov- ery of a tropical model with self-organized criticality (SOC) behavior. Our model is continuous, in contrast to all known models of SOC, and is a cer-

tain scaling limit of the sandpile model, the ﬁrst and archetypical model of SOC. We describe how our model is related to pattern formation and proportional growth phenomena, and discuss the dichotomy between continuous and

1 discrete models in several contexts. Our aim in this context is to present an idealized tropical toy-model (cf. Turing reaction-diffusion model), requiring

further investigation.

Self-Organized Criticality

The statistics concerning earthquakes in a particular extended region of the Earth during a given period of time obey a power law known as the Gutenberg-Richter law (1): the logarithm of the energy of an earthquake is a linear function of the logarithm of the frequency of earthquakes of such energy. An earthquake is an example of a dynamical system presenting both temporal and spatial degrees of freedom. The following deﬁnition generalizes these properties.

Deﬁnition 1. A dynamical system is a Self-Organized Critical (SOC) system if it is slowly driven by an external force, exhibits avalanches, and has power law correlated interactions

(cf. (2), section 7).

By an avalanche, we mean a sudden recurrent modification of the internal energy of the system. In SOC systems, avalanches display scale invariance over energy and over time (2)1. Many ordinary critical phenomena near continuous phase transition (which requires fine tuning) display non-trivial power law correlations with a cut-off associated to the cluster size (4). For example, the Ising model shows power law correlations only for specific parameters, while a self-organized critical system, behaving statistically in an analogous manner, achieves the critical state merely by means of a small external force, without fine-tuning.

1P.W. Anderson describes SOC as having “paradigmatic value” characterizing “the next stage of Physics”. He writes: “In the 21st century, one revolution which can take place is the construction of generalization which jumps and jumbles the hierarchies or generalizations which allow scale-free or scale transcending phenomena. The paradigm for the ﬁrst is broken symmetry; for the second, self-organized criticality” (3).

2 The Sandpile Model

The concept of SOC was introduced in the seminal papers of Per Bak, Chao Tang, and Kurt Wiesenfeld (5,6) where they put forward the archetypical example of a SOC system: the Sand- pile Model. It is a highly idealized cellular automaton designed to display spatio-temporal scale invariance. Unlike other toy models in physics, e.g. the Ising model, a sandpile automaton doesn’t attempt to apprehend the actual interactions of a physical system exhibiting SOC (such as a real sandpile). The sandpile cellular automaton is rather a mathematical model 2 that re- produces the statistical behavior of very diverse dynamical systems: it is descriptive at the level of global emergent phenomena without necessarily corresponding to such systems at the local reductionist level.

0 1 3 4 3 4 0 4 • • • toppling 1 3 1 4 •

Figure 1: Numbers represent the number of sand grains in the vertices of the grid, and a toppling is performed. Red points are unstable vertices.

Imagine a large domain on a checkered plane; each vertex of this grid is determined by two integer coordinates (i, j). We will write Z2 to denote the integral square lattice formed by all pairs of integers (i, j) in the Euclidean plane R2. Our dynamical system will evolve inside Ω, which is the intersection of Z2 with a large convex ﬁgure in the plane. Throughout this paper, we use the word graph to indicate a collection of line segments (called edges) that connect points (called vertices) to each other.

Deﬁnition 2. A sandpile model consists of a grid inside a convex domain Ω on which we place

2A holistic model in the sense of (7).

3 grains of sand at each vertex; the number of grains on the vertex (i, j) is denoted by ϕ(i, j) (see

Figure 1). Formally, a state is an integer-valued function ϕ :Ω → Z≥0. We call a vertex (i, j) unstable whenever there are four or more grains of sand at (i, j), i.e., whenever ϕ(i, j) ≥ 4. The evolution rule is as follows: any unstable vertex (i, j) topples spontaneously by sending one grain of sand to each of its four neighbors (i, j + 1), (i, j − 1), (i − 1, j), (i + 1, j). The sand that falls outside Ω disappears from the system. Stable vertices cannot be toppled. Given an initial state ϕ, we will denote by ϕ◦ the stable state reached after all possible topplings have been performed. It is a remarkable and well-known fact that the ﬁnal state ϕ◦ does not depend on the order of topplings. The ﬁnal state ϕ◦ is called the relaxation of the initial state ϕ.

Bak and his collaborators proposed the following experiment: take any stable state ϕ0 and perturb it at random by adding a grain of sand at a random location. Denote the relaxation of the perturbed state by ϕ1, and repeat this procedure. Thus a sequence of randomly chosen vertices

◦ (ik, jk) gives rise to a sequence of stable states by the rule ϕk+1 = (ϕk + δikjk ) .

3 The relaxation process (ϕk +δikjk ) 7→ ϕk+1 is called an avalanche ; its size is the number of vertices that topple during the relaxation. Given a long enough sequence of uniformly chosen vertices (ik, jk), we can compute the distribution for the sizes of the corresponding avalanches. Let N(s) be the number of avalanches of size s; then the main experimental observation of (5) is that:

log N(s) = τ log s + c.

In other words, the sizes of avalanches satisfy a power law. In Figure ??, we have reproduced this result with τ ∼ −1.2. This has only very recently been given a rigorous mathematical proof using a deep analysis of random trees on the two-dimensional integral lattice Z2 (8). 3We can think of an avalanche as an earthquake.

4 Deﬁnition 3. A recurrent state is a stable state appearing inﬁnitely often, with probability one, in the above dynamical evolution of the sandpile.

Surprisingly, the recurrent states are exactly those which can be obtained as a relaxation of a state ϕ ≥ 3 (pointwise). The set of recurrent states forms an Abelian group (9) and its identity exhibits a fractal structure in the scaling limit (Fig. 6); unfortunately, this fact has resisted a rigorous explanation so far. The main point of this paper is to exhibit fully analogous phenomena in a continuous system (which is not a cellular automaton) within the ﬁeld of tropical geometry.

An advantage of the tropical model is that, while it has self-organized critical behavior, just as the classical model does, its states look much less chaotic; thus we say that the tropical model has no combinatorial explosion.

Mathematical Modeling for Proportional Growth and Pattern Formation

The dichotomy between continuous mathematical models and discrete cellular automata has an important example in developmental biology.

A basic continuous model of pattern formation was offered by Alan Turing in 1952 (10). He suggested that two or more homogeneously distributed chemical substances, termed mor- phogens, with different diffusing rates and chemical activity, can self-organize into spatial patterns of different concentrations. This theory was conﬁrmed decades later and can be applied, for example, to modeling the patterns of ﬁsh and lizard skin (11), (12), or of seashells’ pigmentation (13).

On the discrete modeling side, the most famous model is the Game of Life developed by Conway in 1970 (14). A state of this two-dimensional system consists of ”live” and ”dead” cells that function according to simple rules. Any live cell dies if there are fewer than two or

5 more than three live neighbors. Any dead cell becomes alive if there are three live neighbors. A very careful control of the initial state produces ”oscillators” and ”spaceships” that certainly look fascinating but seem not to be related to realistic biological models. Nevertheless, a strong philosophical conclusion of the Game of Life is that extremely simple rules can produce behavior of arbitrary complexity. A more realistic cellular automaton has recently been derived from the continuous reaction-diffusion model of Turing. In (15), the transformation of the juvenile white ocelli skin patterns of the lizard Timon lepidus into a green and black labyrinth was observed. In this study, the authors presented the skin squamae of lizard as a hexagonal grid, where the color of each individual cell depended on the color states of neighboring positions. This cell automaton could successfully generate patterns coinciding with those on the skin of adult lizards.

6 of Z2 with a large convex figure in the plane. Throughout this Mathematical Modeling for Proportional Growth and paper, we use the word graph to indicate a collection of line Pattern Formation segments (called edges) that connect points (called vertices) The dichotomy between continuous mathematical models and to each other. discrete cellular automata has an important example in devel- Definition 2. A sandpile model consists of a grid inside a opmental biology. convex domain on which we place grains of sand at each A basic continuous model of pattern formation was oered vertex; the number of grains on the vertex (i, j) is denoted by Alan Turing in 1952 (10). He suggested that two or more by Ï(i, j) (see Figure 1). Formally, a state is an integer- homogeneously distributed chemical substances, termed mor- valued function Ï : Z 0. We call a vertex (i, j) unstable phogens, with dierent diusing rates and chemical activity, æ Ø whenever there are four or more grains of sand at (i, j),i.e., can self-organize into spatial patterns of dierent concentra- whenever Ï(i, j) 4. The evolution rule is as follows: any tions. This theory was confirmed decades later and can be Ø unstable vertex (i, j) topples spontaneously by sending one applied, for example, to modeling the patterns of fish and grain of sand to each of its four neighbors (i, j+1), (i, j 1), (i lizard skin (11),(12), or of seashells’ pigmentation (13). ≠ ≠ 1,j), (i +1,j). The sand that falls outside disappears from On the discrete modeling side, the most famous model is the system. Stable vertices cannot be toppled. Given an initial the Game of Life developed by Conway in 1970 (14). A state state Ï, we will denote by Ï¶ the stable state reached after of this two-dimensional system consists of "live" and "dead" all possible topplings have been performed. It is a remarkable cells that function according to simple rules. Any live cell dies and well-known fact that the final state Ï¶ does not depend if there are fewer than two or more than three live neighbors. on the order of topplings. The final state Ï¶ is called the Any dead cell becomes alive if there are three live neighbors. relaxation of the initial state Ï. A very careful control of the initial state produces "oscillators" and "spaceships" that certainly look fascinating but seem not to Bak and his collaborators proposed the following experi- be related to realistic biological models. Nevertheless, a strong ment: take any stable state Ï0 and perturb it at random by philosophical conclusion of the Game of Life is that extremely adding a grain of sand at a random location. Denote the relax- simple rules can produce behavior of arbitrary complexity. A ation of the perturbed state by Ï1, and repeat this procedure. more realistic cellular automaton has recently been derived Thus a sequence of randomly chosen vertices (ik,jk) gives rise from the continuous reaction-diusion model of Turing. In (15), to a sequence of stable states by the rule Ïk+1 =(Ïk + ”ikjk )¶. the transformation of the juvenile white ocelli skin patterns The relaxation process (Ïk + ”i j ) Ïk+1 is called an of the lizard Timon lepidus into a green and black labyrinth k k ‘æ avalanche‡; its size is the number of vertices that topple during was observed. In this study, the authors presented the skin the relaxation. Given a long enough sequence of uniformly squamae of lizard as a hexagonal grid, where the color of each chosen vertices (ik,jk), we can compute the distribution for individual cell depended on the color states of neighboring the sizes of the corresponding avalanches. Let N(s) be the positions. This cell automaton could successfully generate number of avalanches of size s; then the main experimental patterns coinciding with those on the skin of adult lizards. observation of (5) is that: log N(s)=· log s + c.

In other words, the sizes of avalanches satisfy a power law. In Figure 7A, we have reproduced this result with · 1.2. ≥≠ This has only very recently been given a rigorous mathematical proof using a deep analysis of random trees on the two-dimensional integral lattice Z2 (8). (A) (B) (C)

Deﬁnition 3. A recurrent state is a stable state appearing inﬁnitely often, with probability one, in the above dynamical evolution of the sandpile.

Surprisingly, the recurrent states are exactly those which can be obtained as a relaxation of a state Ï 3 (pointwise). Ø The set of recurrent states forms an Abelian group (9) and its (D) (E) (F) identity exhibits a fractal structure in the scaling limit (Fig. 6); Figure 2: In (A), (B), (C) and (D), a very large number N of grains of sand is placed at the Fig. 2. In (A), (B), (C) and (D), a very large number N of grains of sand is placed unfortunately, this fact has resisted a rigorous explanation so origin of the everywhere empty integral lattice, the final relaxed state shows fractal behavior. at the origin of the everywhere empty integral lattice, the final relaxed3 state4 shows Here, as we advance from (A) to (D), we see√ successive sandpiles for N = 10 (A), 10 (B), far. fractal105 (C), behavior. and 106 (D), Here, rescaled as we by factors advance of fromN. In (A) (E), to we (D), zoom we in see on a smallsuccessive region of sandpiles (D) to The main point of this paper is to exhibit fully analogous forshowN = its intricate 103 (A), fractal104 structure,(B), 105 and,(C), finally, and 10 in (F),6 (D), we further rescaled zoom by in factors on a small of Ô portionN. In of (E), we(E). zoom We can in on see a proportional small region growth of (D) occurring to show in the its patterns intricate as thefractal fractal structure, limit appears. and, The finally, phenomena in a continuous system (which is not a cellular balanced graphs inside the roughly triangular regions of (F) are tropical curves. automaton) within the field of tropical geometry. in (F), we further zoom in on a small portion of (E). We can see proportional growth occurring in the patterns as the fractal limit appears. The balanced graphs inside the An advantage of the tropical model is that, while it has roughlyPattern triangular formation regions is related of (F) to an are important tropical problem curves. in developmental biology: how to self-organized critical behavior, just as the classical model explain proportional growth. It is not clear why different parts of animal or human bodies does, its states look much less chaotic; thus we say that the grow at approximately the same rate from birth to adulthood. Sandpile simulations provide a tropical model has no combinatorial explosion. Pattern formation is related to an important problem in developmentalqualitative toy model as biology: follows. how to explain proportional growth. ‡We can think of an avalanche as an earthquake. ItExample is not 4. clearEarly on, why it was di observederent experimentally parts of that animal sandpiles have or humanfractal structure bodies in

2 | www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX 7 Kalinin et al. their spatial degrees of freedom (see Figure 2).

This example exhibits the phenomenon of proportional growth and scale invariance: If we

1 rescale tiles to have area N and let N go to inﬁnity, then the picture converges in the weak-? sense. (See (16) and references therein.) Recently, the patches and a linear pattern in this fractal picture were explained in (17–19) using discrete superharmonic functions and Apollonian circle packing. Dhar and Sadhu (20) established certain two-dimensional sandpile models where the size of newly formed patterns depends on the number of sand grains added on the top, but the number and shape of the patterns remain the same. Strikingly, they also proposed a three-dimensional model which also forms proportionally growing patterns; these patterns look like the head, thorax, and abdomen of real larva. The perspective of our paper suggests that continuous tropical geometry should have consequences in the study of proportional growth modeling. We conjecture that tropical functions should appear as gradients of growth.(Compare Figure 3 in (21) and our Figure 5; see also Section 2.5 in (22)).

Tropical Geometry

Tropicalization can be thought of as the study of geometric and algebraic objects on the log-log scale4 in the limit when the base of the logarithm is very large5. Let us start by considering the most basic mathematical operations: addition and multiplication. With the logarithmic change of coordinates, and with the base of the logarithm becoming inﬁnitely large, multiplication becomes addition and addition becomes taking the maximum. Namely, deﬁne

4Drawing algebraic varieties on log-log scale was successfully used by O. Ya. Viro, in his study of Hilbert’s 17th problem, to construct real algebraic curves with prescribed topology, for a more recent account of this story read (23). 5This limit has been discovered several times in physics and mathematics and was named Maslov dequantiza- tion in mathematical physics before it was called tropicalization in algebraic geometry (24).

8 x y x +t y := logt(t + t )

x y x ×t y := logt(t t ) for x, y positive and then, taking the limit as t tends to +∞, set

Trop(x + y) := lim x +t y = max(x, y), t→+∞

Trop(x × y) := lim x ×t y = x + y. t→+∞

Tropical numbers are, thus, deﬁned as the set of ordinary real numbers (together with {−∞}, i.e. T := R ∪ {−∞}) and with tropical addition max(x, y) and multiplication x + y. The tropical numbers form a semi-ring: all the usual rules of algebra hold in T with the exception of additive inverses (the neutral element for addition is −∞; additive inverses are missing).

This mathematical structure simpliﬁes some computations, and several optimization prob- lems are solved efﬁciently in this way; this is a growing area of applied research in the current decade (it has been used in auctions for the Bank of England (25), for biochemical networks (26), and for Dutch railroads, among other things (27)). Thermodynamics suggests that the tropical limit should be understood as the low temperature limit T → 0 (or equivalently, as the limit when the Boltzmann constant k vanishes, k → 0) of the classical algebraic and geometric operations. In this interpretation of the limit,

− 1 the relevant change of variables is t = e kT , where t is the tropical parameter (the base of the logarithm) and T is the temperature (28–30). More relevant to realistic physical systems, tropical functions and tropical algebra appear naturally in statistical physics as the formal limit as k → 0, and this can be used to analyze frustrated systems like spin ice and spin glasses. This is directly related to the appearance of the tropical degeneration in dimer models (28, 31).

9 Figure 3: Each column represents a tropical degeneration (from top to bottom). The complex picture is at the top; in the middle, we depict the amoeba on log-log scale and, at the bottom, we picture the tropical curve which can be seen as the spine of the corresponding amoeba. In the first example, we have a degree-one curve with three points A, B, C going off to infinity (the intersections A, B of a line with the coordinate axes go to −∞ under the logarithm, and C, the intersection at infinity, goes to (+∞, +∞)). In the second example, a quadric degenerates sending off six points off to infinity. A quadric intersects each of coordinate lines and the line at infinity in two points, therefore, six (three times two) points go off to infinity. From top to bottom t → ∞ 10 Definition 5. Let A ⊂ Z2 be any finite set. A tropical polynomial in two variables is a function

Trop(F (x, y)) = max (aij + ix + jy). (6) (i,j)∈A

A tropical polynomial should be thought of the tropicalization of a classical polynomial

P i j F (x, y) = aijx y obtained by replacing all of the summations and multiplications by (i,j)∈A their tropical counterparts, i.e. max and addition, respectively. For an excellent introduction to tropical geometry, we refer the reader to (32).

Tropical Limit in Algebraic Geometry

Complex algebraic geometry is the ﬁeld of mathematics that studies geometric objects inside complex Euclidean spaces Cn that arise as zero sets of ﬁnite families of polynomials. A simple example of such an object is an algebraic curve C given by a single polynomial equation in two variables

2 X i j C : F (x, y) = 0, (x, y) ∈ C ,F (x, y) = aijx y . (i,j)∈A Here A is a ﬁnite set of pairs of positive integers. The degree of F is the maximal total power i + j of all the monomials xiyj appearing in F , namely d = max (i + j). The curve C is (i,j)∈A a two-dimensional object, despite its name (for two real dimensions is the same as one complex dimension). A non-singular complex curve C = {F (x, y) = 0} of degree d = deg(F ) is a Riemann

1 surface of genus g = 2 (d − 1)(d − 2). Therefore, the usual lines (d = 1) and quadrics (d = 2) are topological 2-spheres, while cubics (d = 3, also called elliptic curves) are tori. The geometric counterpart of the tropicalization is as follows. Given a complex algebraic curve Ct deﬁned by a polynomial

X aij i j Ft(x, y) = γijt x y = 0, |γij| = 1, (i,j)∈A

11 we call the amoeba At the image of Ct under the map logt(x, y) = (logt |x|, logt |y|), At := logt(Ct). The limit of the amoebas At as t → +∞ is called Trop(C), the tropicalization of Ct. The limit Trop(C) can be described entirely in terms of the tropical polynomial Trop(F ) (eq. 6). This fact can be proved by noticing that on the linearity regions of Trop(F (x, y)), one monomial in Ft dominates all the others and, therefore, Ft cannot be zero; and, consequently, we conclude that the limit Trop(C) is precisely the set of points (x, y) in the plane where the (3-dimensional) graph of the function

Trop(F (x, y)) = max (aij + ix + jy) (i,j)∈A is not smooth. This set of points is known as the corner locus of Trop(F (x, y)). For this reason, we deﬁne the tropical plane curve as the corner locus of a tropical polynomial. As depicted in

Figure 3, the tropical degeneration Trop(C) of a complex curve C is essentially the graph of the curve depicted in a logt-logt scale for t very large. It is not very hard to verify that every tropical curve Trop(C) in the plane is a ﬁnite balanced graph, all of whose edges have rational slopes, such that at every vertex, the (weighted) directions of the primitive vectors in every direction cancel out (this is called the balancing condition and is akin to Kirchhoff’s law for electrical circuits). Conversely, every such balanced graph on the plane appears as a tropical algebraic curve Trop(C). Here, it may be a good moment to point out that one can observe small tropical curves in Figure 2, where the ﬁnal state of a sandpile is depicted; this already points towards a relationship between SOC and tropical geometry.

String Theory, Mirror Symmetry

Tropical geometry is intimately related to the interaction between algebraic geometry and string theory that occurs in the mirror symmetry program. Given a tropical 2-dimensional surface B, ˜ we can use the tropical structure to produce a pair (XB, XB) of mirror manifolds. This moti-

12 vated M. Kontsevich to predict that counting tropical curves on B could be used for the calculation of Gromov-Witten invariants (which count holomorphic complex curves) in the A-side of mirror symmetry. The ﬁrst example of such a calculation done from a rigorous mathematical perspective was accomplished by Mikhalkin (33). This perspective has been expanded by Gross and Siebert in their program to understand mirror symmetry from a mathematical viewpoint using tropical geometry (34). Let us also mention that the dichotomy between continuous and discrete models in our paper (already appearing in the biological models) has an important analogue in string theory:

Iqbat et al. have argued that, when we probe space-time beyond the scale α0 and below Planck’s scale, the resulting ﬂuctuations of space-time can be computed with a classical cellular automaton (a melting crystal) representing quantum gravitational foam (35). Their theory is a three-tier system whose levels are classical geometry (Kahler¨ gravity), tropical geometry (toric manifolds) and cellular automata (discrete melting crystals). The theory that we propose in this paper is also a three-tier system whose levels are classical complex algebraic geometry, tropical geometry (analytic tropical curves) and cellular automata (sandpiles). This seems not to be a coincidence and suggests deep connections between our model for SOC and their model for quantum gravitational foam.

Tropical Curves in Sandpiles

To understand the appearance of tropical geometry in sandpiles, consider the toppling function

H(i, j) deﬁned as follows: Given an initial state ϕ and its relaxation ϕ◦, the value of H(i, j) equals the number of times that there was a toppling at the vertex (i, j) in the process of taking ϕ to ϕ◦.

The discrete Laplacian of H is deﬁned by the net ﬂow of sand, ∆H(i, j) :=

H(i − 1, j) + H(i + 1, j) + H(i, j − 1) + H(i, j + 1) − 4H(i, j).

13 The toppling function is clearly non-negative on Ω and vanishes at the boundary. The function ∆H completely determines the ﬁnal state ϕ◦ by the formula:

ϕ◦(i, j) = ϕ(i, j) + ∆H(i, j). (7)

14 tropical wave operators Gp. Given a ﬁxed vertex p =(i0,j0), we deﬁne the wave operator Wp acting on states Ï of the sandpile as:

Wp(Ï):=(Tp(Ï + ”p) ”p)¶, ≠

where Tp is the operator that topples the state Ï+”p at p once, if it’s possible to topple p at all. In a computer simulation, the application of this operator looks like a wave of topplings spreading from p, while each vertex topples at most once. (A) (B) The first important property of Wp is that, for the initial state Ï := 3 + ”P , we can achieve the final state Ï¶ by È Í successive applications of the operator Wp Wp a large 1 ¶ ···¶ r but finite number of times (in spite of the notation):

Ï¶ =(Wp Wp )ŒÏ + ”P . 1 ··· r

This process decomposes the total relaxation Ï Ï¶ into ‘æ layers of controlled avalanching. The second important property of the wave operator Wp is that its action on a state Ï = 3 +f has an interpretation in È Í (C) (D) terms of tropical geometry. To wit, whenever f is a piecewise

Figure 4: The evolution of h3i+δ . Sand falling outside the border disappears. Time progresses linear function with integral slopes that, in a neighborhood of Fig. 4. The evolution of 3 +P ” . Sand falling outside the border disappears. Time in the sequence (A), (B), (C),È andÍ finallyP (D). Before (A), we add grains of sand to several points p, is expressed as ai0j0 + i0x + j0y, we have progressesof the constant in state theh3 sequencei (we see their (A), positions (B), (C), as blue and disks finally given (D). by δ BeforeP ). Avalanches (A), we ensue. add At grains oftime sand (A), to the several avalanches points have barelyof the started. constant At the state end, at3 time(we (D), see we gettheir a tropical positions analytic as blue curve on the square Ω. White represents the region with 3È grainsÍ of sand while green represent disks given by ”P ). Avalanches ensue. At time (A), the avalanches have barely Wp( 3 + f)= 3 + W (f), δ started.2, yellow Atrepresents the end, 1, and at red time represents (D), we the get zero a region. tropical We analytic can think curve of the blue on thedisks squareP as . È Í È Í the genotype of the system, of the state h3i as the nutrient environment, and of the thin graph Whitegiven by represents the tropical thefunction region in (D) with as the 3 grains phenotype of of sand the system. while green represent 2, yellow where W (f) has the same coecients aij as f except one, represents 1, and red represents the zero15 region. We can think of the blue disks ”P as the genotype of the system, of the state 3 as the nutrient environment, and of namely aiÕ j = ai0j0 +1. This is to account for the fact that the È Í 0 0 the thin graph given by the tropical function in (D) as the phenotype of the system. support of the wave is exactly the face where ai0j0 + i0x + j0y is the leading part of f. It can be shown by induction that the toppling function H We will write Gp := WpŒ to denote the operator that satisfies the Least Action Principle:ifÏ(i, j)+F (i, j) 3 is applies Wp to 3 + f until p lies in the corner locus of f.We Æ È Í stable, then F (i, j) H(i, j). Ostojic (36) noticed that H(i, j) repeat that it has an elegant interpretation in terms of tropical Ø is a piecewise quadratic function in the context of Example 1. geometry: Gp increases the coecient ai0j0 corresponding to Consider a state Ï which consists of 3 grains of sand at a neighborhood of p, lifting the plane lying above p in the every vertex, except at a finite family of points graph of f by integral steps until p belongs to the corner locus of Gpf. Thus Gp has the eect of pushing the tropical curve P = p =(i ,j ),...,pr =(ir,jr) towards p (Figure 5) until it contains p. { 1 1 1 } From the properties of the wave operators, it follows imme- where we have 4 grains of sand: diately that:

Œ Ï := 3 + ”p + + ”p = 3 + ”P . [3] FP =(Gp1 Gpr ) 0, È Í 1 ··· r È Í ···

¶ where 0 is the function which is identically zero on .All The state Ï and the evolution of the relaxation can be k intermediate functions (Gp Gp ) 0 are less than H since described by means of tropical geometry (the final picture 1 ··· r (D) of Figure 4 is a tropical curve). This was discovered by they represent partial relaxations, but their limit belongs to P , and this, in turn, implies that H = FP . Caracciolo et al. (37) while a rigorous mathematical theory F to prove this fact has been given by Kalinin et al. (38), which Conclusion. We have shown that the toppling function H we review presently. It is a remarkable fact that, in this case, for Eq. 3 is piecewise linear. Thus, applying Eq. 2, we obtain that Ï¶ is equal to 3 everywhere but the locus where H =0, the toppling function H(i, j) is piecewise linear (after passing ” to the scaling limit). i.e. its corner locus, namely, an -tropical curve (Figure 4). To prove this, one considers the family P of functions F Definition 5. (41)An-tropical series is a piecewise linear on that are: (1) piecewise linear with integral slopes, (2) function on given by: non-negative over and zero at its boundary, (3) concave, and (4) not smooth at every point pi of P .LetFP be the F (x, y)= min (aij + ix + jy), pointwise minimum of functions in P . Then FP H by the (i,j) F Ø œA Least Action Principle (since FP 0, FP (pi) < 0). Æ where the set is not necessarily finite and F ˆ =0.An Lemma 1. In the scaling limit H = FP . A | -tropical curve is the set where F is not smooth. Each - A sketch of a proof. We introduce the wave operators Wp tropical curve is a locally finite graph satisfying the balancing (39, 40) at the cellular automaton level and the corresponding condition.

Kalinin et al. PNAS | June 21, 2018 | vol. XXX | no. XX | 5 It can be shown by induction that the toppling function H satisﬁes the Least Action Princi- ple: if ϕ(i, j) + ∆F (i, j) ≤ 3 is stable, then F (i, j) ≥ H(i, j). Ostojic (36) noticed that H(i, j) is a piecewise quadratic function in the context of Example 4. Consider a state ϕ which consists of 3 grains of sand at every vertex, except at a ﬁnite family of points

P = {p1 = (i1, j1), . . . , pr = (ir, jr)} where we have 4 grains of sand:

ϕ := h3i + δp1 + ··· + δpr = h3i + δP . (8)

The state ϕ◦ and the evolution of the relaxation can be described by means of tropical geometry (the ﬁnal picture (D) of Figure 4 is a tropical curve). This was discovered by Caracciolo et al. (37) while a rigorous mathematical theory to prove this fact has been given by Kalinin et al. (38), which we review presently. It is a remarkable fact that, in this case, the toppling function H(i, j) is piecewise linear (after passing to the scaling limit).

To prove this, one considers the family FP of functions on Ω that are: (1) piecewise linear with integral slopes, (2) non-negative over Ω and zero at its boundary, (3) concave, and (4) not smooth at every point pi of P . Let FP be the pointwise minimum of functions in FP . Then

FP ≥ H by the Least Action Principle (since ∆FP ≤ 0, ∆FP (pi) < 0).

Lemma 9. In the scaling limit H = FP .

A sketch of a proof. We introduce the wave operators Wp (39,40) at the cellular automaton level and the corresponding tropical wave operators Gp. Given a ﬁxed vertex p = (i0, j0), we deﬁne the wave operator Wp acting on states ϕ of the sandpile as:

◦ Wp(ϕ) := (Tp(ϕ + δp) − δp) ,

16 where Tp is the operator that topples the state ϕ + δp at p once, if it’s possible to topple p at all. In a computer simulation, the application of this operator looks like a wave of topplings spreading from p, while each vertex topples at most once.

The ﬁrst important property of Wp is that, for the initial state ϕ := h3i + δP , we can achieve

◦ the ﬁnal state ϕ by successive applications of the operator Wp1 ◦ · · · ◦ Wpr a large but ﬁnite number of times (in spite of the notation):

◦ ∞ ϕ = (Wp1 ··· Wpr ) ϕ + δP .

This process decomposes the total relaxation ϕ 7→ ϕ◦ into layers of controlled avalanching.

The second important property of the wave operator Wp is that its action on a state ϕ = h3i + ∆f has an interpretation in terms of tropical geometry. To wit, whenever f is a piecewise

linear function with integral slopes that, in a neighborhood of p, is expressed as ai0j0 +i0x+j0y, we have

Wp(h3i + ∆f) = h3i + ∆W (f),

0 where W (f) has the same coefﬁcients aij as f except one, namely ai0j0 = ai0j0 + 1. This is to account for the fact that the support of the wave is exactly the face where ai0j0 + i0x + j0y is the leading part of f.

∞ We will write Gp := Wp to denote the operator that applies Wp to h3i+∆f until p lies in the corner locus of f.We repeat that it has an elegant interpretation in terms of tropical geometry:

Gp increases the coefﬁcient ai0j0 corresponding to a neighborhood of p, lifting the plane lying above p in the graph of f by integral steps until p belongs to the corner locus of Gpf. Thus Gp has the effect of pushing the tropical curve towards p (Figure 5) until it contains p. From the properties of the wave operators, it follows immediately that:

∞ FP = (Gp1 ··· Gpr ) 0,

17 k where 0 is the function which is identically zero on Ω. All intermediate functions (Gp1 ··· Gpr ) 0 are less than H since they represent partial relaxations, but their limit belongs to FP , and this, in turn, implies that H = FP . Conclusion. We have shown that the toppling function H for Eq. 8 is piecewise linear. Thus, applying Eq. 7, we obtain that ϕ◦ is equal to 3 everywhere but the locus where ∆H 6= 0, i.e. its corner locus, namely, an Ω-tropical curve (Figure 4).

Deﬁnition 10. (41) An Ω-tropical series is a piecewise linear function on Ω given by:

F (x, y) = min (aij + ix + jy), (i,j)∈A where the set A is not necessarily ﬁnite and F |∂Ω = 0. An Ω-tropical curve is the set where F is not smooth. Each Ω-tropical curve is a locally ﬁnite graph satisfying the balancing condition.

18 Wp p• p•

(A) (B)

FigureFig. 5. 5: Top:Top: The The action action of the of thewave wave operator operatorWp on aW tropicalp on a curve. tropical The curve.tropical curveThe tropical steps p W p closercurve to stepsby an closer integral to p step.by an Thus integralp shrinks step. the Thus faceW thatp shrinksbelongs the to; face the combinatorial that p belongs p G 0 morphologyto; the combinatorial of the face that morphologybelongs ofto, the can face actually that change.p belongs Bottom: to, can The actually function change.p , where p is the center of the circle, and its associated omega-tropical curve are shown. Bottom: The function Gp0, where p is the center of the circle, and its associated omega-tropical curve are shown. Remark 11. Tropical curves consist of edges, such that to each direction of the edges there corresponds a line-shaped pattern (a string) such as the one encountered in Figure 2; these patternsRemark can be computed 1. Tropical (18). In simulations, curves we consist have observed of thatedges, these strings such act that like the to (C) (D) renormalizationeach direction group and, of thus, the ensure edges the proportional there corresponds growth of the quadratic a line-shaped patches in Fig- pattern (a string) such as the one encountered in Figure 2; Fig. 6. The first two pictures show the comparison between the classical (A) and ure 2. The same occurs in other sandpile models with proportional growth, which suggests that tropical (B) sandpiles for P = 100 generic points on the square. In (C), the square these patterns can be computed (18). In simulations, we have | | has side N = 1000; a large number ( P = 40000) of grains has been added, tropical geometry is a less reductionist tool than cellular automata to study this phenomenon. | | observed that these strings act like the renormalization group showing the spatial SOC behavior on the tropical model compared to the identity and, thus, ensure the proportional growth of the quadratic (D) of the sandpile group on the square of side N = 1000. In the central square patches in Figure 2. The same19 occurs in other sandpile models region on (C) (corresponding to the solid block of the otherwise fractal unit), we have a random tropical curve with edges on the directions (1, 0), (0, 1), and ( 1, 1), which with proportional growth, which suggests that tropical geome- ± is given by a small perturbation of the coefficients of the tropical polynomial defining try is a less reductionist tool than cellular automata to study the usual square grid. this phenomenon.

We will call the modification Fk 1 Fk the k-th avalanche. ≠ æ It occurs as follows: To the state Fk 1 we apply the tropical The Tropical Sandpile Model ≠ operators Gp1 ,Gp2 ,...,Gpk ; Gp1 ,... in cyclic order until the function stops changing; the discreteness of the coordinates of Here, we define a new model, the tropical sandpile (TS), reflect- the points in P ensures that this process is finite. Again, as ing structural changes when a sandpile evolves. The definition before, while sandpile-inspired, the operators Gp are defined of this dynamical system is inspired by the mathematics of entirely in terms of tropical geometry without mention of the previous section; TS is not a cellular automaton but it sandpiles. exhibits SOC. There is a dichotomy: Each application of an operator The dynamical system lives on the convex set =[0,N] ◊ Gp, either does something to change the shape of the current [0 N]; we will consider to be a very large square. The input tropical curve (in this case Gp is called an active operator), or data of the system is a large but finite collection of points does nothing, leaving the curve intact (if p already belongs to P = p ,...,pr with integer coordinates on the square . { 1 } the curve). Each state of the system is an -tropical series (and the associated -tropical curve). Definition 6. The size of the k-th avalanche is the number of distinct active operators Gpi used to take the system from The initial state for the dynamical system is F0 = 0, and its the self-critical state Fk 1 to the next self-critical state Fk, final state is the function FP defined previously. Notice that ≠ divided by k. In particular, the size sk of the k-th avalanche is the definition of P , while inspired by sandpile theory, uses F a number between zero and one, 0 sk 1, and it estimates no sandpiles or cellular automata whatsoever. Intermediate Æ Æ the proportional area of the picture that changed during the states Fk k ,...,r satisfy the property that Fk is not smooth { } =1 avalanche. at p1,p2,...,pk; i.e. the corresponding tropical curve passes through these points. In the previous example, as the number of points in P In other words, the tropical curve is first attracted to the grows and becomes comparable to the number of lattice points point p1. Once it manages to pass through p1 for the first in , the tropical sandpile exhibits a phase transition going time, it continues to try to pass through p ,p . Once it into spatial SOC (fractality). This provides the first evidence { 1 2} manages to pass through p ,p , it proceeds in the same in favor of SOC on the tropical sandpile model, but there is { 1 2} manner towards p ,p ,p . This process is repeated until { 1 2 3} If the coordinates of the points in P are not integers, the model is well-defined, but we need to take the curve passes through all of P = p ,...,pr . a limit (see (41)), which is not suitable for computer simulations. { 1 }

6 | www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX Kalinin et al. The Tropical Sandpile Model

Here, we define a new model, the tropical sandpile (TS), reflecting structural changes when a sandpile evolves. The definition of this dynamical system is inspired by the mathematics of the previous section; TS is not a cellular automaton but it exhibits SOC. The dynamical system lives on the convex set Ω = [0,N] × [0 N]; we will consider Ω to be a very large square. The input data of the system is a large but finite collection of points

P = {p1, . . . , pr} with integer coordinates on the square Ω. Each state of the system is an Ω-tropical series (and the associated Ω-tropical curve).

The initial state for the dynamical system is F0 = 0, and its final state is the function FP defined previously. Notice that the definition of FP , while inspired by sandpile theory, uses no sandpiles or cellular automata whatsoever. Intermediate states {Fk}k=1,...,r satisfy the property that Fk is not smooth at p1, p2, . . . , pk; i.e. the corresponding tropical curve passes through these points.

In other words, the tropical curve is ﬁrst attracted to the point p1. Once it manages to pass through p1 for the ﬁrst time, it continues to try to pass through {p1, p2}. Once it manages to pass through {p1, p2}, it proceeds in the same manner towards {p1, p2, p3}. This process is repeated until the curve passes through all of P = {p1, . . . , pr}.

20 Wp p• p•

(A) (B)

Fig. 5. Top: The action of the wave operator Wp on a tropical curve. The tropical curve steps closer to p by an integral step. Thus Wp shrinks the face that p belongs to; the combinatorial morphology of the face that p belongs to, can actually change. Bottom: The function Gp0, where p is the center of the circle, and its associated omega-tropical curve are shown.

Remark 1. Tropical curves consist of edges, such that to (C) (D) each direction of the edges there corresponds a line-shaped Figure 6: The first two pictures show the comparison between the classical (A) and tropical (B) pattern (a string) such as the one encountered in Figure 2; sandpilesFig. 6. forThe|P first| = 100 twogeneric pictures points show on the square. comparison In (C), thebetween square Ω thehas classical side N = (A) 1000 and; atropical large number (B) sandpiles (|P | = 40000 for P) of= grains 100 hasgeneric been pointsadded,on showing the square. the spatial In (C), SOC the behavior square these patterns can be computed (18). In simulations, we have | | on thehas tropical side N model= 1000 compared; a large to the number identity (D) ( P of= the 40000 sandpile) of group grains on the has square been of added, side | | observed that these strings act like the renormalization group Nshowing= 1000. theIn the spatial central SOC square behavior region on (C) on (correspondingthe tropical model to the solid compared block of to the the otherwise identity and, thus, ensure the proportional growth of the quadratic fractal(D) of unit), the sandpile we have a group random on tropical the square curve with of side edgesN on= the 1000 directions. In the(1, central0), (0, 1) square, and (±1, 1), which is given by a small perturbation of the coefficients of the tropical polynomial region on (C) (corresponding to the solid block of the otherwise fractal unit), we have a patches in Figure 2. The same occurs in other sandpile models defining the usual square grid. random tropical curve with edges on the21 directions (1, 0), (0, 1), and ( 1, 1), which with proportional growth, which suggests that tropical geome- ± is given by a small perturbation of the coefficients of the tropical polynomial defining try is a less reductionist tool than cellular automata to study the usual square grid. this phenomenon.

6 | www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX Kalinin et al. We will call the modiﬁcation Fk−1 → Fk the k-th avalanche. It occurs as follows: To the

state Fk−1 we apply the tropical operators Gp1 ,Gp2 ,...,Gpk ; Gp1 ,... in cyclic order until the function stops changing; the discreteness of the coordinates of the points in P ensures that

6 this process is ﬁnite . Again, as before, while sandpile-inspired, the operators Gp are deﬁned entirely in terms of tropical geometry without mention of sandpiles.

There is a dichotomy: Each application of an operator Gp, either does something to change the shape of the current tropical curve (in this case Gp is called an active operator), or does nothing, leaving the curve intact (if p already belongs to the curve).

Deﬁnition 12. The size of the k-th avalanche is the number of distinct active operators Gpi used to take the system from the self-critical state Fk−1 to the next self-critical state Fk, divided by k.

In particular, the size sk of the k-th avalanche is a number between zero and one, 0 ≤ sk ≤ 1, and it estimates the proportional area of the picture that changed during the avalanche.

In the previous example, as the number of points in P grows and becomes comparable to the number of lattice points in Ω, the tropical sandpile exhibits a phase transition going into spatial SOC (fractality). This provides the first evidence in favor of SOC on the tropical sandpile model, but there is a far more subtle spatio-temporal SOC behavior that we will exhibit in the following paragraphs. While the ordering of the points from the 1st to the r-th is important for the specific details of the evolution of the system, the system’s statistical behavior and final state are insensitive to it. This is called an Abelian property, which was studied extensively in (42) for discrete dynamical systems (Abelian networks). Our model suggests studying continuous dynamical systems with this Abelian property, such as, for example, Abelian networks with nodes on the plane, a continuous set of states, and an evolution rule depending on the coordinates of the

6If the coordinates of the points in P are not integers, the model is well-deﬁned, but we need to take a limit (see (41)), which is not suitable for computer simulations.

22 nodes. We expect that the Abelian property is equivalent to the least action principle (cf. (42)).

Self-Organized Criticality in the Tropical World

The tropical sandpile dynamics exhibit slow driving avalanching (in the sense of (2) page 22). Once the tropical dynamical system stops after r steps, we can ask ourselves what the statistical behavior of the number N(s) of avalanches of size s is like. We posit that the tropical dynamical system exhibits spatio-temporal SOC behavior; that is, we have a power law:

log N(s) = τ log s + c.

To conﬁrm this, we have performed experiments in the supercomputing clusters ABACUS and Xiuhcoatl at Cinvestav (Mexico City); the code is available on (43). In the ﬁgure below, we see the graph of log N(s) vs. log s for the tropical (piecewise linear, continuous) sandpile dynamical system; the resulting experimental τ in this case was τ ∼ −0.9.

Conclusion and Further Directions

We have obtained a piecewise-linear (continuous, tropical) model to study statistical aspects of non-linear phenomena. As tropical geometry is highly developed, it seems reasonable to believe that it will provide new reductionist explanations for physical non-linear systems in the future (as well as providing a tool for studying proportional growth phenomena in biology and elsewhere), as has already happened with non-linear aspects of algebraic geometry and mirror symmetry. Next, we list open questions.

From the point of view of real physical phenomena, the tropical sandpile provides a new class of mathematical and modelling tools. Some possible directions for further research are as follows:

23 Direction 1. If Ω is a polygon with sides of rational slope, then each Ω-tropical curve C can be obtained as the corner locus of FP for a certain set P of points. To achieve this, one considers a (ﬁnite) set P of points which all belong to C and cover C rather densely, meaning that every point in C is very close to a point in P (at a distance less than a very small ). As is shown in (41), the corner locus of FP is an Ω-tropical curve passing though P which solves a certain Steiner-type problem: minimizing the tropical symplectic area. However, because the set P is huge in this construction, an open question remains: Can we ﬁnd a small set P = {p1, . . . , pr} (where r is approximately equal to the number of faces of C) such that C is the corner locus of

FP ? (We thank an anonymous referee for this question.)

24 a far more subtle spatio-temporal SOC behavior that we will exhibit in the following paragraphs. While the ordering of the points from the 1st to the r-th is important for the speciﬁc details of the evolution of the system, the system’s statistical behavior and ﬁnal state are insensitive to it. This is called an Abelian property, which was studied 10000 y=-1.2x+1.47 extensively in (42) for discrete dynamical systems (Abelian networks). Our model suggests studying continuous dynamical

systems with this Abelian property, such as, for example, 1000 Abelian networks with nodes on the plane, a continuous set of states, and an evolution rule depending on the coordinates of the nodes. We expect that the Abelian property is equivalent 100

to the least action principle (cf. (42)). log(frequency)

Self-Organized Criticality in the Tropical World 10 The tropical sandpile dynamics exhibit slow driving avalanching (in the sense of (2) page 22). Once the tropical dynamical system stops after r steps, we can ask ourselves what the statistical behavior of the number 1

N(s) of avalanches of size s is like. We posit that the tropical 0.01 0.02 0.05 0.10 0.20 0.50 1.00 dynamical system exhibits spatio-temporal SOC behavior; that is, we have a power law: log(avalanche)

log N(s)=· log s + c. (A) Sandpile dynamical system

To conﬁrm this, we have performed experiments in the supercomputing clusters ABACUS and Xiuhcoatl at Cinvestav (Mexico City); the code is available on (43). In the ﬁgure below, we see the graph of log N(s) vs. log s for the tropical 10000 (piecewise linear, continuous) sandpile dynamical system; the y=-0.9x+2.2 resulting experimental · in this case was · 0.9. ≥≠

Conclusion and Further Directions 1000 We have obtained a piecewise-linear (continuous, tropical) model to study statistical aspects of non-linear phenomena. As tropical geometry is highly developed, it seems reasonable 100 to believe that it will provide new reductionist explanations for log(frequency) physical non-linear systems in the future (as well as providing

a tool for studying proportional growth phenomena in biol- 10 ogy and elsewhere), as has already happened with non-linear aspects of algebraic geometry and mirror symmetry. Next, we list open questions. From the point of view of real physical phenomena, the 1 tropical sandpile provides a new class of mathematical and 0.01 0.02 0.05 0.10 0.20 0.50 1.00 modelling tools. Some possible directions for further research are as follows: log(s)

Direction 1. If is a polygon with sides of rational slope, (B) Tropical dynamical system then each -tropical curve C can be obtained as the corner Fig. 7. A) The power law for sandpiles. The logarithm of the frequency is linear locus of FP for a certain set P of points. To achieve this, one considers a (finite) set P of pointsFigure which all 7: belong A) The to C powerand with law respect for to sandpiles. the logarithm of the The avalanche logarithm size, except of near the the right frequency where the is linear with respect avalanches are larger than half of the system. Here we have =[0, 100]2 and cover C rather densely, meaning thatto the every logarithm point in C is of very the avalancheis initially filled with size,3 grains except everywhere, near followed the by right106 dropped where grains. the B) avalanchesThe are larger than close to a point in P (at a distancehalf less of than the a verysystem. small Here‘). power-law we have for theΩ tropical = (piece-wise [0, 100] linear,2 and continuous) is initially dynamical system.filled In with this 3 grains everywhere, F computer experiment has a side of 1000 units and we add at random a set P of As is shown in (41), the corner locus of P is an -tropical6 curve passing though P which solvesfollowed a certain by Steiner-type10 dropped10000 grains.individual B) sand The grains (apower-law random large genotype). for the tropical (piece-wise linear, continu- problem: minimizing the tropicalous) symplectic dynamical area. However, system. In this computer experiment Ω has a side of 1000 units and we add at because the set P is huge in this construction, an open question Direction 2. The operators Gp can be lifted to the algebraic remains: Can we find a small set Prandom= p ,...,p a setr (whereP of 10000r is settingindividual but, for now, sand we grains can do this (a random only in fields large of characteris- genotype). { 1 } approximately equal to the number of faces of C) such that C tic two (41). Is it possible to lift Gp in characteristic zero (the is the corner locus of FP ? (We thank an anonymous referee complex realm)? In any case, we expect that this diculty can for this question.) be alleviated by a mirror symmetry interpretation. The issue

Kalinin et al. PNAS | June25 21, 2018 | vol. XXX | no. XX | 7 Direction 2. The operators Gp can be lifted to the algebraic setting but, for now, we can do this only in ﬁelds of characteristic two (41). Is it possible to lift Gp in characteristic zero (the complex realm)? In any case, we expect that this difﬁculty can be alleviated by a mirror symmetry interpretation. The issue is due to the symplectic nature of Gp: indeed, Ω-tropical curves naturally appear as tropical symplectic degenerations (44, 45). When Ω is not a rational polygon, one should take into account non-commutative geometry (46). What is the mirror

∗ analog of Gp in the complex world? We expect that there should exist an operator Gp acting on strings, and through this we expect power-law statistics for a mirror notion of the areas of the faces of Ω-tropical curves. Closely related to this, in analogy to the work of Iqbar et al. (35), we conjecture that the partition function obtained by summing over statistical mechanical conﬁgurations of sandpiles should have, via mirror symmetry, an interpretation as a path integral in terms of Kahler¨ geometry. Tropical geometry should play the role of toric geometry in this case. What is the precise geometric model in this situation?

Developments in this direction would allow a new renormalization group interpretation of SOC (cf. (47, 48)).

Direction 3. Rastelli and Kanel¨ studied nanometer sized three-dimensional islands formed during epitaxial growth of semiconductors appearing as faceted pyramids that seem to be modeled by tropical series from an inspection of Figure 5 in this paper and Figure 3 in (21). It would be very interesting to prove that this is so and to study the consequences of this observation to the modeling of the morphology of such phenomena. This may not be totally unrelated to allometry in biology. Is it possible that the gradient slope model, as in Section 2.5 in (22), could be piecewise linear, and the corner locus slopes could prescribe the type and speed of growth for tissues?

Direction 4. Study the statistical distributions for the coefﬁcients of the tropical series in Figure

26 6 (C) in our paper. Explain why the slopes are mostly of directions (0, 1), (1, 0), (1, 1), (−1, 1). Is it possible that a concentration measure phenomenon takes place and that such a type of picture appears with probability one? (We thank Lionel Levine for this question.)

Experimental Data

The data used to produce Figure 9 can be found in (43).

Acknowledgments

Nikita Kalinin was funded by the SNSF PostDoc.Mobility Grant 168647, supported in part by Young Russian Mathematics award, and would like to thank Grant FORDECYT-265667

”Programa para un Avance Global e Integrado de la Matematica´ Mexicana”. Also, support from the Basic Research Program of the National Research University Higher School of Economics is gratefully acknowledged.

Yulieth Prieto was funded by Grant FORDECYT-265667 and by ABACUS (Cinvestav). Mikhail Shkolnikov was supported by ISTFELLOW program. Finally, Ernesto Lupercio would like to thank the Moshinsky Foundation, Conacyt, FORDECYT-

265667, ABACUS, Xiuhcoatl, IMATE-UNAM, Samuel Gitler International Collaboration Cen- ter and the Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. No. 14.641.31.0001 and the kind hospitality of the University of Geneva and of the Mathematisches Forschungsin- stitut Oberwolfach where this work started. In memoriam JL.

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