Relaxation in one-dimensional tropical sandpile

Abstract

A relaxation in the tropical sandpile model is a process of deforming a tropical hypersurface towards a finite collection of points. We show that, in the one-dimensional case, a relaxation terminates after a finite number of steps. We present experimental evidence suggesting that the number of such steps obeys a power law.

Figures (7)

• Figure 1: A plot of the tropical series $z\mapsto\inf_{v\in\mathbb{Z}^2\backslash\{0\}} (|v|+z\cdot v)$ on the unit disk.
• Figure 2: Two-dimensional illustration of the geometric interpretation of the operator $G_p$ seen as a result of a continuous process that shrinks a face to which the point $p$ belongs.
• Figure 3: One-dimensional illustration of the operator $G_p.$ Plots of $F$ and $G_pF$ are shown in bold, the slopes of linear pieces are written near them in gray. In this example, the original hypersurface $H$ defined by $F$ contains a point of multiplicity two, since there is an extra monomial (its plot is shown in gray) in the canonical form of $F$ contributing at this point, or, equivalently, since the slope jumps by two at this point. Applying $G_p$ at the level of tropical polynomials means increasing the coefficient of the constant term (moving upwards the horizontal segment of the plot in this example) until the function has a break at $p.$ At the level of sets of non-linearity of tropical polynomials, $G_p$ moves two points towards $p$ until at least one of them coincides with $p.$
• Figure 4: Relaxation for points $p=4$ and $q=3$ on $\Omega=[0,9],$ represented by hypersurfaces (denoted by collections of circles -- two concentric circles circle mean a point of multiplicity two) defined by $\Omega$-tropical polynomials $0_\Omega,$$G_p 0_\Omega,$$G_qG_p 0_\Omega,$$G_p G_qG_p 0_\Omega,$ etc. The lowest picture is stable, i.e. the points $p$ and $q$ belong to the hypersurface defined by the result of the relaxation.
• Figure 5: Numerical approximations for loci of $(p,q)\in(0,1)^2$ with length of relaxation $L(p,q)$ equal to $1,2,3,4,5$ or $6$ (left to right, up to down). The pictures are made in R.

Theorems & Definitions (3)

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