Tropical Limits of Decimated Polynomials

Motivated by some problems that originate in Statistical Physics and Algebraic Dynamics, we discuss a particular renormalization mechanism of multivariate Laurent polynomials which is called a decimation, and the corresponding tropical limiting shape result obtained in Arzhakova et al. (Decimation of principal actions. Preprint, 2018).


Introduction
Young students attending the evening mathematical seminar at the Moscow State School no. 57, will most probably encounter the following problem: To Rafail Kalmanovich Gordin, on the occasion of his 70th birthday.  Is it possible to tile the 8 × 8 chessboard without two opposite corners with 2 × 1 dominoes?
A simple parity argument shows that it is, indeed, not possible. A more difficult question is in how many ways we can tile the full 8 × 8, and, more generally, 2n × 2n chessboard with the 2 × 1 dominos.
In 1961, a Dutch physicist Piet Kasteleyn found complete solutions of several 'arrangement problems' of such nature (Kasteleyn 1963). In particular, he showed that the number of domino tilings of a chessboard of a size 2n × 2n is given by Z n = n−1 m=0 n−1 k=0 4 − 2 cos 2m + 1 2n + 1 π − 2 cos 2k + 1 2n + 1 π . (1.1) There is a one-to-one correspondence between the domino tilings and the dimer configurations, or, in other words, the perfect matchings of a corresponding graph, as Fig. 1 demonstrates. A perfect matching of a graph is a subset of its edges such that every vertex of the graph is incident to exactly one edge of the subset.
The method developed by Kasteleyn is not only applicable to counting the number of domino tilings, or equivalently, dimer configurations, but can also be used to compute the weighted sums of the form: where M(G 2n×2n ) is a collection of all dimer configurations of the square box of size 2n × 2n. For any dimer configuration (matching) M, its weight is given by Obtaining explicit expressions for the partition functions Z n (1.2) is important in Statistical Physics, as it allows the computation of the free energy F(u, v) given by and, by analysing the free energy, one is able to determine some important macroscopic properties of the systems in the thermodynamic limit. In particular, the singularities of the free energy function indicate the presence of the phase transitions. It turns out (Cohn et al. 2001;, that it is easier to compute a weighted partition function of the form (1.2) if we embed the 2n × 2n square grid on a torus (see Fig. 2).
The partition function Z T n for weighted dimer matchings on a torus possesses an elegant explicit expression: where f is a Laurent polynomial in two variables x, y, namely, and for every integer n ≥ 1, n−1 k=0 f e 2πi m n x, e 2πi k n y . (1.5) The limiting free energy F does not depend on whether we consider the partition function Z n or Z T n . In both cases, More generally, expressions of a form (1.4) and (1.6) hold for all planar simple bipartite Z 2 -periodic graphs and some associated polynomials f . For example, for the honeycomb lattice, f = a + bx + cy, where a, b, c are the weights of the horizontal, north-east, and south-east edges, respectively.
Okounkov, Kenyon, and Sheffield [see ] obtained a beautiful limiting shape result for the coefficients of f (n) (x, y) in (1.5) for the special 'dimer' polynomials f . In the present note we discuss the generalization of the result of Okounkov, Kenyon, and Sheffield to arbitrary Laurent polynomials in d variables, where d ≥ 2.

Decimation of Polynomials
A Laurent polynomial in d commuting variables z 1 ,…,z d , can be presented as a sum: where we use the multi-index notation m = (m 1 , . . . , m d ). The sum has a finite number of terms: there are only finitely many m's with f m = 0. The multi-indices m ∈ Z d with f m = 0 are called the exponent vectors. The set of all exponent vectors is called the support of f and is denoted by supp( f ). Now, fix an arbitrary integer n ≥ 1. Then, the nth decimation of a Laurent polynomial f (z 1 , . . . , z d ) is defined as: ( 2.1) Our interest in these polynomials arose when studying decimations (renormalization transformation) of the so-called principle algebraic actions-a natural class of algebraic dynamics, see Arzhakova et al. (2018) for more details. Such polynomials have been considered earlier by Purbhoo (2008) who studied approximations of amoebas.In the present paper, we will discuss properties of the decimated polynomials.
For every n, a decimated polynomial f (n) is again a Laurent polynomial Moreover, the resulting exponent vectors of f (n) are entry-wise divisible by n. In other words, f (n) is a Laurent polynomial in z n 1 , . . . , z n d .

Example 1
Consider a polynomial f (x, y) = 1 − x − y. Then, the first three decimations are (see Fig. 3): We remind the reader that the Newton polytope N ( f ) of a Laurent polynomial f (z 1 , . . . , z d ) is a subset of R d which is a convex hull of the exponent vectors of f (z 1 , . . . , z d ). Note that the Newton polytopes of f (n) and of f satisfy the following relation: Therefore, when n increases, the Newton polytope of f (n) grows, and so do the absolute values of the non-zero coefficients of f (n) m . In fact, their growth rate is exponential in n. The natural question is whether there is a scaling limit of the coefficients of f (n) . Namely, whether the limits , it was shown that for special 'dimer' polynomials in 2 variables, the limits (2.2) do exist. However, the method of  is not applicable to general Laurent polynomials as it relies on the physical and combinatorial properties of a two-dimensional model of dimer matchings. The polynomials that arise in this model form a subset in the set of two-dimensional Laurent polynomials and possess certain physical interpretation which allows to show that the associated surface tension is strictly convex (Sheffield 2005). The convexity of the surface tension implies the existence of the desired limit; however, a generic Laurent polynomial does not allow a 'dimer' interpretation and, thus, cannot be analysed using the methods of . However, the tropical geometry provides a convenient framework to address the limit questions in general case.
Tropical geometry incorporates many facets of algebraic geometry and convex analysis (Maclagan and Sturmfels 2015).
Let us consider a Laurent polynomial f where ·, · denotes the standard scalar product on R d . Tropicalization of f is thus a tropical analogue of the Laurent polynomial m log | f m |z m .
The tropical variety of f (z) is the set of all points t ∈ R d such that the maximum in (3.1) is achieved at at least two monomials. Therefore, the tropicalization of f is a piecewise affine convex function on R d ; each component of the complement of the tropical variety defines a domain where a certain monomial of f is maximal, c.f. (3.1). Figure 4 shows the tropical varieties of the first 4 decimations of a polynomial 1 + x + y.
In a joint work with Doug Lind and Klaus Schmidt (Arzhakova et al. 2018), we established the following result on the existence of scaling limits of tropicalizations of the decimated polynomials f (n) .

namely, R f (n) (t) = n d R f (t).
Hence, it suffices to compare trop( f (n) ) and R f . The Ronkin function can be easily bounded from above: indeed, let us denote by supp( f (n) ) the number of integer points inside the support of f (n) . Then, Hence, for all n ∈ Z and t ∈ R d one has Since the cardinality of the support of f (n) grows at most as const·n d , we immediately conclude that Let us start the discussion of the lower bound of R f (t) with the following observation. Suppose that z = (z 1 , . . . , z d ) ∈ (C * ) d is a d-tuple of non-zero complex numbers; denote by t j ∈ R and φ j ∈ T ∼ [0, 1) the modulus and the argument of z j for every j = 1, . . . , d, i.e., z j = e t j +2πiφ j . Note that × f e t 1 +2πiφ 1 +2πik 1 /n , . . . , e t d +2πiφ d +2πik d /n . (3.5) The expression on the right hand side is a Riemann sum for the integral (3.3) defining the Ronkin function R f (t). Note also, that despite the fact f may have zeros on a torus {z : |z| = e t }, the function log | f | is still integrable since the singularities are only logarithmic. One naturally expects that for almost all z, the Riemann sums in (3.5) converge to R f (t). However, establishing such convergence turns out to be a rather intricate Diophantine problem (Dimitrov 2016;Lind et al. 2013). Fortunately, in order to establish the lower bound, one does not have to deal with a convergence problem in a complete generality. It suffices to prove that the Riemann sums are bounded from above by R f (t).
We say that the initial value z = (e t 1 +2πiφ 1 , . . . , e t d +2πiφ d ) is good if the points of the set Q n (z) = e t 1 +2πiφ 1 +2πik 1 /n , . . . , e t d +2πiφ d +2πik d /n , k 1 , . . . k d = 0, . . . n − 1 do not fall or come too close (depending on n) to the variety of f : V f = {z ∈ (C * ) d : f (z) = 0}. For good points, it is relatively easy to show that the Riemann sums are close to the value of the integral R f (t). On the contrary, for the 'bad' initial values z, the points in Q n (z), which are close to V f , give a negative contribution to the sum (3.5). Hence, one is able to conclude that for all z with |z 1 | = e t 1 , . . . , |z d | = e t d , (3.6) or, equivalently, The final part of the argument is based on a relatively simple statement from the Fourier analysis: if the absolute value of a complex (trigonometric) polynomial is bounded from above by some constant M on a torus T t = {z : |z| = e t } then the absolute values of all of its monomials are also bounded from above by the same constant. Therefore, applying this result to f (n) and the inequality (3.7), we conclude that that for all m and, hence, Combining the inequalities (3.4) and (3.8), we obtain the desired result.
Remark 1 Theorem 1 provides some insight on the geometry of tropical varieties of f (n) . In Fig. 4, the similarity between the shapes of tropical varieties of decimations of f = 1 + x + y for various n is not accidental. Let us recall the notion of an amoeba of a Laurent polynomial f of d variables that was first suggested by Gelfand, Kapranov, and Zelevinsky in Gelfand et al. (2008). An amoeba of f denoted by A f is an image of the variety V f under the map Log : V f → R d given by the formula Log(z 1 , . . . , z d ) = (log |z 1 |, . . . , log |z d |).
The amoeba A f is a closed subset of R d with a non-empty convex complement. The Ronkin function R f is strictly convex over A f and affine on each component of R d \A f (Mikhalkin 2004) (Fig. 5).
It is easy to see that A f coincides with A f (n) for every positive n. According to Theorem 1, 1 n d trop( f (n) )(t) is a sequence of piecewise-affine convex functions converging to the Ronkin function R f which is affine on the complement of A f and strictly convex inside A f . Therefore, the outer boundary of the tropical varieties 1 n d trop( f (n) ) converge to the boundary of A f as n approaches infinity.

Surface Tension
The Legendre transform (or a dual) of a function F : R d → R is defined as The Legendre transform F * is always a convex function; moreover, for a convex closed function F, the Legendre transform is an involution: Suppose f is a Laurent polynomial. Then, the tropicalization of f is, in fact, a Legendre transform of the function F defined as follows: Indeed, one has Applying the Legendre transform once again, we obtain F * * = trop( f ) * . Since F, in general, is not convex, F * * = F. However, F * * is easy to describe; namely, F * * = conv(F), where conv(F) is the so-called greatest convex minorant or a convex hull of F: the largest convex function satisfying conv(F)(u) ≤ F(u) for all u. Clearly, conv(F)(u) = + ∞ for al u / ∈ N ( f ) and is finite on N ( f ).
Using the above arguments for the polynomials f (n) and the result of the Theorem 1, one immediately obtains the following result: Corollary 1 Let conv(F (n) ) be the greatest convex minorant of Then, for all u ∈ R d , By analogy with , we refer to the function σ f as to the surface tension of f .
Corollary 1 should be seen as a weaker, but at the same time, a more general version of the result established in  for polynomials appearing in dimer problems. It turns out that these polynomials are rather special in the following sense: for such polynomials, one is able to define the surface tension using the coefficients of f (n) without the need to resort to convex hulls of the coefficients. In other words, some form of convexity is already present in the coefficients of f (n) . It is, of course, very interesting to identify such polynomials.  showed that for every Harnack curve, one can construct a polynomial of 2 variables, whose algebraic variety is the given Harnack curve, and for which the surface tension is well defined. At the present moment, it is not clear which conditions could play a similar role in higher dimensions.
Finally, we would like to remark that the use of tropical geometric methods to study limits of partition functions or similar quantities is very natural, and has been proposed in Itenberg and Mikhalkin (2012).
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