Counting Arithmetical Structures Luis David Garc´ ıa Puente Department of Mathematics and Statistics Sam Houston State University Blackwell-Tapia Conference 2018 The Institute for Computational and Experimental Research in Mathematics Providence, RI
Statistical physics, a matrix and an abelian group Gutenberg-Richter Law (1956) The relationship between the magnitude M and total number N of earthquakes in any given region and time period of magnitude ≥ M is N ∝ 10 − bM , where b is a constant ∼ 1 .
Statistical physics, a matrix and an abelian group Gutenberg-Richter Law (1956) The relationship between the magnitude M and total number N of earthquakes in any given region and time period of magnitude ≥ M is N ∝ 10 − bM , where b is a constant ∼ 1 . For each earthquake with magnitude M ≥ 4 there are about ◮ 0.1 with M ≥ 5 ◮ 0.01 with M ≥ 6 ◮ . . .
Gutenberg-Richter Law (in terms of Energy) Shallow worldwide earthquakes 1976 - 2005 (Global Centroid Moment Tensor Project)
Gutenberg-Richter Law (in terms of Energy) Shallow worldwide earthquakes 1977-2010 (Global Centroid Moment Tensor Project)
Gutenberg-Richter Law Per Bak (1996) “This law is amazing! How can the dynamics of all the elements of a system as complicated as the crust of the earth, with mountains, valleys, lakes, and geological structures of enormous diversity, conspire, as by magic, to produce a law with such extreme simplicity?”
Self-organization towards criticality Self-organized criticality (SOC) is a property of dynamical systems that have a critical point as an attractor. [Bak, Tang, Wiesenfeld (1987)] Their macroscopic behavior displays the scale-invariance characteristic of the critical point of a phase transition, but without the need to tune control parameters to precise values. This property is considered to be one of the mechanisms by which complexity arises in nature. It has been extensively studied in the statistical physics literature during the last three decades.
Mathematical Model for Sandpiles In 1987, Bak, Tang, and Wiesenfeld proposed the following model that captures important features of self-organized criticality. ◮ The model is defined on a rectangular grid of cells. ◮ The system evolves in discrete time. ◮ At each time step a sand grain is dropped onto a random grid cell. ◮ When a cell amasses four grains of sand, it becomes unstable . ◮ It relaxes by toppling whereby four sand grains leave the site, and each of the four neighboring sites gets one grain. ◮ This process continues until all sites are stable .
Mathematical Model for Sandpiles In 1987, Bak, Tang, and Wiesenfeld proposed the following model that captures important features of self-organized criticality. ◮ Start this process on an empty grid. ◮ At first there is little activity. ◮ As time goes on, the size (the total number of topplings performed) of the avalanche caused by a single grain of sand becomes hard to predict. ◮ A recurrent (or critical) configuration is a stable configuration that appears infinitely often in this Markov process.
Distribution of sandpiles Distribution of avalanche sizes in a 10 × 10 grid.
Abelian Sandpile Model on Graphs [Dhar (1990)] v 1 v 2 Let Γ = ( V , E , s ) denote a finite, con- s nected, loopless multigraph with a dis- tinguished vertex s called the sink . v 4 v 3 2 4 A configuration over Γ is a function s σ : V \ { s } − → N . 0 1
Stable Configurations Given a graph Γ = ( V , E , s ), for each v ∈ V \ { s } let d v be the number of edges incident to the vertex v .
Stable Configurations Given a graph Γ = ( V , E , s ), for each v ∈ V \ { s } let d v be the number of edges incident to the vertex v . Definition A configuration c is stable if and only if c ( v ) < d v for all v ∈ V \ { s } .
Stable Configurations Given a graph Γ = ( V , E , s ), for each v ∈ V \ { s } let d v be the number of edges incident to the vertex v . Definition A configuration c is stable if and only if c ( v ) < d v for all v ∈ V \ { s } . Toppling 3 4 s 0 3
Stable Configurations Given a graph Γ = ( V , E , s ), for each v ∈ V \ { s } let d v be the number of edges incident to the vertex v . Definition A configuration c is stable if and only if c ( v ) < d v for all v ∈ V \ { s } . Toppling 3 4 4 0 s s 0 3 0 4
Stable Configurations Given a graph Γ = ( V , E , s ), for each v ∈ V \ { s } let d v be the number of edges incident to the vertex v . Definition A configuration c is stable if and only if c ( v ) < d v for all v ∈ V \ { s } . Toppling 3 4 4 0 0 2 s s s 0 3 0 4 2 0
Sandpile Groups Theorem (Dhar (1990)) Given a graph Γ = ( V , E , s ) . The set S (Γ) of recurrent sandpiles together with stable addition forms a finite Abelian group, called the sandpile group of Γ .
Laplacian of a Graph Given a graph Γ = ( V , E , s ) with n + 1 vertices. The reduced Laplacian � L of Γ is the n × n matrix defined by d i if i = j � L ij = − m ij if ij ∈ E , with multiplicity m ij 0 otherwise where we assume that the sink s is the ( n + 1)-st vertex.
Laplacian of a Graph Given a graph Γ = ( V , E , s ) with n + 1 vertices. The reduced Laplacian � L of Γ is the n × n matrix defined by d i if i = j � L ij = − m ij if ij ∈ E , with multiplicity m ij 0 otherwise where we assume that the sink s is the ( n + 1)-st vertex. v 1 v 2 4 − 1 0 − 1 s − 1 4 − 1 0 � L = . 0 − 1 4 − 1 − 1 0 − 1 4 v 4 v 3
Invariant Factors of the Sandpile Group S (Γ) Theorem (Dhar (1990)) Given a graph Γ = ( V , E , s ) with reduced Laplacian � L. Let diag ( k 1 , k 2 , . . . , k n ) be the Smith Normal Form of � L. Then the sandpile group S (Γ) is isomorphic to S (Γ) ∼ = Z k 1 × Z k 2 × · · · × Z k n .
Invariant Factors of the Sandpile Group S (Γ) Theorem (Dhar (1990)) Given a graph Γ = ( V , E , s ) with reduced Laplacian � L. Let diag ( k 1 , k 2 , . . . , k n ) be the Smith Normal Form of � L. Then the sandpile group S (Γ) is isomorphic to S (Γ) ∼ = Z k 1 × Z k 2 × · · · × Z k n . v 1 v 2 4 − 1 0 − 1 − 1 4 − 1 0 � s L = . 0 − 1 4 − 1 − 1 0 − 1 4 v 4 v 3
Invariant Factors of the Sandpile Group S (Γ) Theorem (Dhar (1990)) Given a graph Γ = ( V , E , s ) with reduced Laplacian � L. Let diag ( k 1 , k 2 , . . . , k n ) be the Smith Normal Form of � L. Then the sandpile group S (Γ) is isomorphic to S (Γ) ∼ = Z k 1 × Z k 2 × · · · × Z k n . v 1 v 2 1 0 0 0 0 1 0 0 � L = . s 0 0 8 0 0 0 0 24 v 4 v 3
Invariant Factors of the Sandpile Group S (Γ) Theorem (Dhar (1990)) Given a graph Γ = ( V , E , s ) with reduced Laplacian � L. Let diag ( k 1 , k 2 , . . . , k n ) be the Smith Normal Form of � L. Then the sandpile group S (Γ) is isomorphic to S (Γ) ∼ = Z k 1 × Z k 2 × · · · × Z k n . v 1 v 2 S (Γ) ∼ s = Z 8 × Z 24 , | S (Γ) | = 8 · 24 = 192 . v 4 v 3
Invariant Factors of the Sandpile Group S (Γ) Theorem (Dhar (1990)) Given a graph Γ = ( V , E , s ) with reduced Laplacian � L. Let diag ( k 1 , k 2 , . . . , k n ) be the Smith Normal Form of � L. Then the sandpile group S (Γ) is isomorphic to S (Γ) ∼ = Z k 1 × Z k 2 × · · · × Z k n . Theorem (Matrix-Tree Theorem) If Γ is a connected graph, then the number of spanning trees of Γ , denoted κ (Γ) , is equal to the determinant of the reduced Laplacian matrix � L of Γ . So | S (Γ) | = κ (Γ) .
Sandpiles on a grid ◮ The sandpile group of the 2 × 2-grid is Z 8 × Z 24 . ◮ The sandpile group of the 3 × 3-grid is Z 4 × Z 112 × Z 224 . ◮ The sandpile group of the 4 × 4-grid is Z 2 8 × Z 1320 × Z 6600 .
Sandpiles on a grid ◮ The sandpile group of the 2 × 2-grid is Z 8 × Z 24 . ◮ The sandpile group of the 3 × 3-grid is Z 4 × Z 112 × Z 224 . ◮ The sandpile group of the 4 × 4-grid is Z 2 8 × Z 1320 × Z 6600 .
Sandpiles on a grid ◮ The sandpile group of the 2 × 2-grid is Z 8 × Z 24 . ◮ The sandpile group of the 3 × 3-grid is Z 4 × Z 112 × Z 224 . ◮ The sandpile group of the 4 × 4-grid is Z 2 8 × Z 1320 × Z 6600 . Open Problem 1 Find a general formula for the sandpile group of an n × m -grid. Open Problem 2 Give a complete characterization of the identity.
Identity sandpile in the 3276 × 3276 square grid Color scheme: black=0, yellow=1, blue=2, and red=3.
Arithmetical Structures ◮ Let G be a finite, simple, connected graph with n ≥ 2 vertices.
Arithmetical Structures ◮ Let G be a finite, simple, connected graph with n ≥ 2 vertices. ◮ A be the adjacency matrix of G ( A ij = # edges from vertex v i to v j .)
Arithmetical Structures ◮ Let G be a finite, simple, connected graph with n ≥ 2 vertices. ◮ A be the adjacency matrix of G ( A ij = # edges from vertex v i to v j .) ◮ An arithmetical structure of G is a pair ( d , r ) ∈ Z n > 0 × Z n > 0 such that r is primitive (gcd of its coefficients = 1) and (diag( d ) − A ) r = 0 .
Recommend
More recommend
