arXiv subjects

Mikhail Shkolnikov

Publications and source records attributed to Mikhail Shkolnikov.

At least 19 recordsLinked to original sources

arXiv2026

The 2-Adic Valuation of the Order of the All-Ones Class in the Sandpile Group of a Square

Place one grain at every nonsink vertex of the wired $n\times n$ square, and let $L(n)$ be the order of this operation in the sandpile group. Thus $L(n)$ is the least positive $q$ for which $q$ uniform grain layers form an integral combination of toppling moves. We prove that, for every $n\ge1$, \[ \nu_2(L(n))= \begin{cases} 2,&n=1,\\ 1,&n\ge2\text{ even},\\ \nu_2(n+1)+2,&n\ge3\text{ odd}. \end{cases} \] For even squares, this follows from the domino--sandpile results of Florescu, Morar, Perkinson, Salter, and Xu, completed by a short parity observation. For odd squares, a unimodular cyclic basis identifies the folded cokernel with a quotient by two shifted Chebyshev polynomials and sends the all-ones class to $1$. Its order is determined by the constant part of this polynomial ideal, not just by a determinant. Two normalized Euclidean remainders reduce to consecutive Fibonacci polynomials over $\mathbb F_2$, giving the exact valuation.

math.CO↗

arXiv2026

Symmetry Emergence in Self-Organized Criticality

We describe a mechanism of affine symmetry emergence in the maximal density regime of the prototypical model of self-organized criticality when the inverse square of the mesh of the underlying lattice is much larger than the number of random perturbation points distributed according to a prescribed probability measure supported in the interior of the ambient convex domain. Moreover, an appropriate scaling limit of the toppling function (aka odometer), which counts the number of operations per site, is a solution to a non-linear partial differential equation well known in the context of optimal transport and differential geometry, making it possible to accurately estimate the deviation of the density from its maximal value in any macroscopic window. The mechanism for the affine symmetry emergence is due to the novel empirical fact, supported in addition by inductive arguments that have recently being upgraded to a rigorous proof, that the scaling limit of the toppling function is the unique concave solution of the Monge-Amp\`ere equation with Dirichlet boundary condition on the convex domain with the potential given by the probability measure used above as the infinite-perturbation profile.

math-ph↗

arXiv2026

Many-point tropical relaxation and the Monge--Amp\`ere equation

We prove a quantitative tropical approximation to the planar Aleksandrov Monge--Amp\`ere equation. Let $\Omega\subset\mathbb R^2$ be a bounded open convex domain, fix $K\Subset\Omega$, and let $F_N=G_{P_N}(0_\Omega)$ be the minimal nonnegative concave tropical series with integral slopes, zero boundary values, and corner locus containing a universally generic $N$-point set $P_N\subset K$. Set $u_N=N^{-1/2}F_N$ and $\mu_N=N^{-1}\sum_{p\in P_N}\delta_p$. For every compact $L\Subset\Omega$ we prove $\left|\int\varphi\,d(\mathrm{MA}(u_N)-\mu_N)\right|\le C(\Omega,K,L)N^{-1/2}(\|\varphi\|_\infty+\|\nabla\varphi\|_\infty)$ for $\varphi\in C_c^1(\Omega)$ supported in $L$. If $\mu_N\rightharpoonup\mu$, where $\mu$ is a probability measure supported in $K$, then $u_N$ converges uniformly on $\overline\Omega$ to the unique continuous concave zero-boundary Aleksandrov solution of $\mathrm{MA}(F)=\mu$, and $\mathrm{MA}(u_N)\rightharpoonup\mu$ vaguely in $\Omega$. No regularity or strict convexity of $\partial\Omega$ is assumed. For bounded rational convex polygons, strong genericity suffices. If $P\subset K$ is strongly generic with $|P|=N$ and $F_P=G_P(0_\Omega)$, its tropical curve has exactly $N$ bounded cells; the duals of the uncut marked carriers form a spanning tree; every compact internal edge has weight one; and $\mathrm{MA}(F_P)(\Omega^\circ)=N-1+\tfrac12D_{\mathrm{term}}(F_P)$, with $D_{\mathrm{term}}(F_P)=O_{\Omega,K}(\sqrt N)$. For strongly generic sequences satisfying the same empirical-measure hypothesis, the normalized curvature measures converge weakly on the closed polygon. We also obtain almost-sure limits for i.i.d. samples from absolutely continuous laws supported in $K$, affine covariance of the continuum solution, and, for source sequences covered by the polygonal theorem, a configuration-dependent Abelian-sandpile diagonal.

math.AP↗

arXiv2026

Residues of a tropical zeta function for convex domains

We define an $\operatorname{SL}_n(\mathbb{Z})$-invariant tropical zeta function of a convex domain. In dimension 2 it admits boundary Dirichlet-series representation with summands indexed by Farey pairs. For $C^3$ strictly convex domains, it extends meromorphically to $\Re(s)>3/5$, holomorphic there except for a simple pole at $s=2/3$, with residue universally proportional to equiaffine perimeter. A Tauberian argument yields the $t^{1/3}$ wave-front lattice-perimeter asymptotic for $t\rightarrow 0$. In addition, for a special domain $L$, which is a limit shape of lattice polygons in a square, with its tropical zeta function being expressed in terms of Witten SU(3) zeta function, we compute the exact coefficient in the asymptotic expansion of the integer-averaged lattice point counting for the leading term $N^{1/2}$.

math.NT↗

arXiv2026

Introduction to non-Abelian Patchworking

The note introduces a novel concept of non-Abelian patchworking arising as real locus of non-Abelian complex-phase tropical hypersurfaces, the theory of which is now developed enough to allow the proposed spin-off. Although, non-Abelian Tropical Geometry makes sense for an arbitrary reductive complex group, the state of the art is that of full understanding of tropicalizations of surfaces within three dimensional groups $PGL_2(\mathbb{C})$ and $SL_2(\mathbb{C}),$ which are closely related via the two-fold covering. We stress our point, that this is an announcement of a framework, taking care of explaining explicitly the input, which is more geometric and less combinatorial than in the original Viro's method, to construct possible types of real algebraic surfaces in the real projective 3-space, and verify that it reproduces all the existing isotopy types of surfaces up to degree three. We obtain two general theorems concerning the topology of primitive PGL2 surfaces, observing in particular that they may have different Euler charteristic for a fixed degree greater than one, not necessarily equal to the signature of the corresponding complex surface, which would be the case for primitive combinatorial patchworking due to a result of Itenberg.

math.AG↗

arXiv2026

Fibers of phase tropicalizations

The subject of the present paper is phase tropicalization, which was used crucially in the context of Mikhalkin's correspondence theorem for curve counting in the complex coefficient case. The subject can be traced back to Viro's patchworking for constructing topological types of real algebraic curves. These two instances correspond to complex and real phases. Both fall into the category of what can be called "abelian" or classical tropicalization, referring to degenerations of varieties within an algebraic torus (or its compactification). In contrast, in "non-abelian" tropicalizations the ambient torus is replaced by a non-commutative group such as the special linear group. This is the beginning of a general theory valid for a wide array of coefficient systems and dimensions. As an application, the paper settles the question of phase tropicalization for the special linear group $\mathrm{SL}_2$. It also gives an algebraic explanation and phase extension of the case of curves, previously studied in the purely geometric framework. To accomplish these tasks we introduce valuative tools that allow us to prove an affine version of Kapranov's theorem on tropical hypersurfaces and its generalization to arbitrary tropical varieties. Most notably, we show the functorial properties of the graded ring of a valuation and exhibit the polynomial structure of the graded ring of monomial valuations.

math.AG↗

arXiv2025

Limits of Equi-Affine Equi-Distant Loci of Planar Convex Domains with Two Non-Parallel Asymptotes

In this note, we introduce equi-affine invariants by averaging over the space of tropical structures of fixed covolume. Applied to the tropical distance series, this construction produces a family of equi-affine invariant functions associated with convex domains which are expected to satisfy a number of remarkable properties. We conjecture a limiting description of the associated level sets in the compact case, and we prove an analogue of this statement for unbounded domains with two non-parallel asymptotes, showing the universality of the affine curvature of the resulting limiting hyperbola. In addition, we give an explicit formula for the arithmetic mean value at the center of the unit disk.

math-ph↗

arXiv2025

Algebraic Limits of Sandpiles

The paper contributes to building algebraic foundations of self-organized criticality answering a previously unsolved question about the limiting structure of the extended sandpile group as well as relating it to another limit at the level of classical sandpile groups with respect to certain monomorphisms, and puts forward a concept of canonical sandpile epimorphisms, drawing an unexpected consequence about the divisibility properties of the numbers of spanning trees on rectangles.

math-ph↗

arXiv2025

Tropical limit of hyperbolic amoebas of complex analytic surfaces

In this letter, we establish a general fact about the convergence of images of families of closed analytic surfaces in the special linear group $\operatorname{SL}_2(\mathbb{C})$ under the quotient by its maximal compact subgroup $\operatorname{SU}(2)$ subject to a contracting scaling sequence.

math.AG↗

arXiv2025

Planar tropical caustics: trivalency and convexity

Tropical caustic of a convex domain on the plane is a canonically associated tropical analytic curve inside the domain. In this note we give a graphical proof for the classification of its intermediate vertices, implying in particular that they are always trivalent. Apart from that we explain how various known examples of tropical caustics are constructed and discuss the possibility of relaxing the convexity condition for the domain.

math.AG↗

arXiv2025

$PSL_2$ tropicalization and lines on surfaces

The paper is based on a talk given by the first author at the G\"okova Geometry \& Topology conference in May 2024. The subject is an interplay between the ideas of tropical geometry and two-by-two matrices with an intention to explore new types of geometries. More concretely, the article gives a preliminary account for a non-abelian version of phase tropicalization for subvarieties of $PSL_2.$

math.AG↗

arXiv2025

Introduction to $PSL_2$ phase tropicalization

The usual approach to tropical geometry is via degeneration of amoebas of algebraic subvarieties of an algebraic torus $(\mathbb{C}^*)^n$. An amoeba is logarithmic projection of the variety forgetting the angular part of coordinates, called the phase. Similar degeneration can be performed without ignoring the phase. The limit then is called phase tropical variety, and it is a powerful tool in numerous areas. In the article is described a non-commutative version of phase tropicalization in the simplest case of the matrix group $PSL_2(\mathbb{C})$, replacing here $(\mathbb{C}^*)^n$ in the classical approach.

math.AG↗

arXiv2024

Tropical formulae for summation over a part of SL(2, Z)

Let $f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}$, let $(a,b,c,d)$ stand for $a,b,c,d\in\mathbb Z_{\geq 0}$ such that $ad-bc=1$. Define \begin{equation} \label{eq_main} F(s) = \sum_{(a,b,c,d)} f(a,b,c,d)^s. \end{equation} In other words, we consider the sum of the powers of the triangle inequality defects for the lattice parallelograms (in the first quadrant) of area one. We prove that $F(s)$ converges when $s>1$ and diverges at $s=1/2$. (This papers differs from its published version: Fedor Petrov showed us how to easily prove that $F(s)$ converges for $s>2/3$ and diverges for $s\leq 2/3$, see below.) We also prove $$\sum\limits_{\substack{(a,b,c,d), 1\leq a\leq b, 1\leq c\leq d}} \frac{1}{(a+b)^2(c+d)^2(a+b+c+d)^2} = 1/24,$$ and show a general method to obtain such formulae. The method comes from the consideration of the tropical analogue of the caustic curves, whose moduli give a complete set of continuous invariants on the space of convex domains.

math.NT↗

arXiv2024

Wave fronts and caustics in the tropical plane

The paper studies intrinsic geometry in the tropical plane. Tropical structure in the real affine $n$-space is determined by the integer tangent vectors. Tropical isomorphisms are affine transformations preserving the integer lattice of the tangent space, they may be identified with the group $\operatorname{GL_n}(\mathbb{Z})$ extended by arbitrary real translations. This geometric structure allows one to define wave front propagation for boundaries of convex domains. Interestingly enough, an arbitrary compact convex domain in the tropical plane evolves to a finite polygon after an arbitrarily small time. The caustic of a wave front evolution is a tropical analytic curve. The paper studies geometry of the tropical wave fronts and caustics. In particular, we relate the caustic of a tropical angle to the continued fraction expression of its slope, and treat it as a tropical trigonometry notion.

math.AG↗

arXiv2023

Tropical curves in sandpile models

A sandpile is a cellular automaton on a graph that evolves by the following toppling rule: if the number of grains at a vertex is at least its valency, then this vertex sends one grain to each of its neighbors. In the study of pattern formation in sandpiles on large subgraphs of the standard square lattice, S. Caracciolo, G. Paoletti, and A. Sportiello experimentally observed that the result of the relaxation of a small perturbation of the maximal stable state contains a clear visible thin balanced graph formed by its deviation (less than maximum) set. Such graphs are known as tropical curves. During the early stage of our research, we have noticed that these tropical curves are approximately scale-invariant, that is the deviation set mimics an extremal tropical curve depending on the domain on the plane and the positions of the perturbation points, but not on the mesh of the lattice. In this paper, we rigorously formulate these two facts in the form of a scaling limit theorem and prove it. We rely on the theory of tropical analytic series, which is used to describe the global features of the sandpile dynamic, and on the theory of smoothings of discrete superharmonic functions, which handles local questions.

math.CO↗

arXiv2023

Relaxation in one-dimensional tropical sandpile

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.

math.CO↗

arXiv2021

Non-commutative amoebas

The group of isometries of the hyperbolic 3-space is one of the simplest non-commutative complex Lie groups. Its quotient by the maximal compact subgroup naturally maps it back to the hyperbolic space. Each fiber of this map is diffeomorphic to the real projective 3-space. The resulting map can be viewed as the simplest non-commutative counterpart of the amoeba map introduced, in the commutative setting, by Gelfand, Kapranov and Zelevinsky. The paper surveys basic properties of the non-commutative amoebas and compares them against their commutative counterparts.

math.CV↗

arXiv2019

Sandpile monomorphisms and limits

We introduce a tiling problem between bounded open convex polyforms $\hat{P}\subset\mathbb{R}^2$ with directed and uniquely colored edges. If there exists a tiling of the polyform $\hat{P}_2$ by $\hat{P}_1$, we show that one can construct a monomorphism from the sandpile group $G_{Γ_1}=\mathbb{Z}^{Γ_1}/Δ(\mathbb{Z}^{Γ_1})$ on the domain (graph) $Γ_1=\hat{P}_1\cap\mathbb{Z}^2$ to the respective group on $Γ_2=\hat{P}_2\cap\mathbb{Z}^2$. We provide several examples of infinite series of such tilings with polyforms converging to $\mathbb{R}^2$, and thus the first definition of scaling-limits for the sandpile group on the plane. Additional results include an exact sequence relating sandpile configurations to harmonic functions, an alternative formula for the order of the sandpile group based on a basis for the module of integer-valued harmonic functions, and three examples of how to prove the existence of (cyclic) subgroups for infinite families of sandpile groups by constructing appropriate integer-valued harmonic functions. The main open question concerns if the scaling-limits of the sandpile group for different sequences of polyforms converging to $\mathbb{R}^2$ are isomorphic.

math-ph↗