Counting Arithmetical Structures Luis David Garc a Puente - - PowerPoint PPT Presentation

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

• f 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)]

Let Γ = (V , E, s) denote a finite, con- nected, loopless multigraph with a dis- tinguished vertex s called the sink. v1 s v2 v3 v4 A configuration over Γ is a function σ : V \ {s} − → N. 2 s 4 1

Stable Configurations

Given a graph Γ = (V , E, s), for each v ∈ V \ {s} let dv 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 dv be the number of edges incident to the vertex v .

Definition

A configuration c is stable if and only if c(v) < dv for all v ∈ V \ {s}.

Stable Configurations

Given a graph Γ = (V , E, s), for each v ∈ V \ {s} let dv be the number of edges incident to the vertex v .

Definition

A configuration c is stable if and only if c(v) < dv for all v ∈ V \ {s}.

Toppling

3 s 4 3

Stable Configurations

Given a graph Γ = (V , E, s), for each v ∈ V \ {s} let dv be the number of edges incident to the vertex v .

Definition

A configuration c is stable if and only if c(v) < dv for all v ∈ V \ {s}.

Toppling

3 s 4 3 4 s 4

Stable Configurations

Given a graph Γ = (V , E, s), for each v ∈ V \ {s} let dv be the number of edges incident to the vertex v .

Definition

A configuration c is stable if and only if c(v) < dv for all v ∈ V \ {s}.

Toppling

3 s 4 3 4 s 4 s 2 2

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

• Lij =

     di if i = j −mij if ij ∈ E, with multiplicity mij

• therwise

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

• Lij =

     di if i = j −mij if ij ∈ E, with multiplicity mij

• therwise

where we assume that the sink s is the (n + 1)-st vertex. v1 s v2 v3 v4

• L =

    4 −1 −1 −1 4 −1 −1 4 −1 −1 −1 4     .

Invariant Factors of the Sandpile Group S(Γ)

Theorem (Dhar (1990))

Given a graph Γ = (V , E, s) with reduced Laplacian

• L. Let

diag(k1, k2, . . . , kn) be the Smith Normal Form of

• L. Then the

sandpile group S(Γ) is isomorphic to S(Γ) ∼ = Zk1 × Zk2 × · · · × Zkn.

Invariant Factors of the Sandpile Group S(Γ)

Theorem (Dhar (1990))

Given a graph Γ = (V , E, s) with reduced Laplacian

• L. Let

diag(k1, k2, . . . , kn) be the Smith Normal Form of

• L. Then the

sandpile group S(Γ) is isomorphic to S(Γ) ∼ = Zk1 × Zk2 × · · · × Zkn. v1 s v2 v3 v4

• L =

    4 −1 −1 −1 4 −1 −1 4 −1 −1 −1 4     .

Invariant Factors of the Sandpile Group S(Γ)

Theorem (Dhar (1990))

Given a graph Γ = (V , E, s) with reduced Laplacian

• L. Let

diag(k1, k2, . . . , kn) be the Smith Normal Form of

• L. Then the

sandpile group S(Γ) is isomorphic to S(Γ) ∼ = Zk1 × Zk2 × · · · × Zkn. v1 s v2 v3 v4

• L =

    1 1 8 24     .

Invariant Factors of the Sandpile Group S(Γ)

Theorem (Dhar (1990))

Given a graph Γ = (V , E, s) with reduced Laplacian

• L. Let

diag(k1, k2, . . . , kn) be the Smith Normal Form of

• L. Then the

sandpile group S(Γ) is isomorphic to S(Γ) ∼ = Zk1 × Zk2 × · · · × Zkn. v1 s v2 v3 v4 S(Γ) ∼ = Z8 × Z24, |S(Γ)| = 8 · 24 = 192.

Invariant Factors of the Sandpile Group S(Γ)

Theorem (Dhar (1990))

Given a graph Γ = (V , E, s) with reduced Laplacian

• L. Let

diag(k1, k2, . . . , kn) be the Smith Normal Form of

• L. Then the

sandpile group S(Γ) is isomorphic to S(Γ) ∼ = Zk1 × Zk2 × · · · × Zkn.

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 Z8 × Z24. ◮ The sandpile group of the 3×3-grid is Z4 × Z112 × Z224. ◮ The sandpile group of the 4×4-grid is Z2

8 × Z1320 × Z6600.

Sandpiles on a grid

◮ The sandpile group of the 2×2-grid is Z8 × Z24. ◮ The sandpile group of the 3×3-grid is Z4 × Z112 × Z224. ◮ The sandpile group of the 4×4-grid is Z2

8 × Z1320 × Z6600.

Sandpiles on a grid

◮ The sandpile group of the 2×2-grid is Z8 × Z24. ◮ The sandpile group of the 3×3-grid is Z4 × Z112 × Z224. ◮ The sandpile group of the 4×4-grid is Z2

8 × Z1320 × Z6600.

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 (Aij = # edges from vertex vi to vj.)

Arithmetical Structures

◮ Let G be a finite, simple, connected graph with n ≥ 2 vertices. ◮ A be the adjacency matrix of G (Aij = # edges from vertex vi to vj.) ◮ An arithmetical structure of G is a pair (d, r) ∈ Zn

>0 × Zn >0

such that r is primitive (gcd of its coefficients = 1) and (diag(d) − A) r = 0.

Arithmetical Structures

◮ Let G be a finite, simple, connected graph with n ≥ 2 vertices. ◮ A be the adjacency matrix of G (Aij = # edges from vertex vi to vj.) ◮ An arithmetical structure of G is a pair (d, r) ∈ Zn

>0 × Zn >0

such that r is primitive (gcd of its coefficients = 1) and (diag(d) − A) r = 0. ◮ Let D be the vertex-degree vector of G. Then diag(D) − A is the Laplacian matrix of G and (D, ¯ 1) is an arithmetical structure.

Arithmetical Structures

Proposition (Lorenzini (1989))

The generalized Laplacian matrix L(G, d) := diag(d) − A has rank n − 1, and is an (almost non-singular) M-matrix: Every (proper) principal minor of M is positive.

Corollary

d and r determines each other uniquely.

Arithmetical Structures

◮ Arith(G) := {(d, r) | (d, r) is an arithmetical structure on G}.

Arithmetical Structures

◮ Arith(G) := {(d, r) | (d, r) is an arithmetical structure on G}. ◮ (G, d, r) is called an arithmetical graph.

Arithmetical Structures

◮ Arith(G) := {(d, r) | (d, r) is an arithmetical structure on G}. ◮ (G, d, r) is called an arithmetical graph. ◮ coker L(G, d) = Zn/ im L(G, d) ∼ = Z ⊕ K(G, d, r).

Arithmetical Structures

◮ Arith(G) := {(d, r) | (d, r) is an arithmetical structure on G}. ◮ (G, d, r) is called an arithmetical graph. ◮ coker L(G, d) = Zn/ im L(G, d) ∼ = Z ⊕ K(G, d, r). ◮ K(G, d, r) is a finite abelian group called the critical group of (G, d, r).

Arithmetical Structures

◮ Arith(G) := {(d, r) | (d, r) is an arithmetical structure on G}. ◮ (G, d, r) is called an arithmetical graph. ◮ coker L(G, d) = Zn/ im L(G, d) ∼ = Z ⊕ K(G, d, r). ◮ K(G, d, r) is a finite abelian group called the critical group of (G, d, r). ◮ K(G, D, ¯ 1) is the sandpile group of G.

Arithmetical Structures

◮ Arith(G) := {(d, r) | (d, r) is an arithmetical structure on G}. ◮ (G, d, r) is called an arithmetical graph. ◮ coker L(G, d) = Zn/ im L(G, d) ∼ = Z ⊕ K(G, d, r). ◮ K(G, d, r) is a finite abelian group called the critical group of (G, d, r). ◮ K(G, D, ¯ 1) is the sandpile group of G.

Theorem (Lorenzini ’89)

Arith(G) is finite. (Note: Proof is non-constructive.)

Motivation: Arithmetic Geometry (Lorenzini ’89)

◮ Let C be an algebraic curve that degenerates into n components C1, . . . , Cn.

Motivation: Arithmetic Geometry (Lorenzini ’89)

◮ Let C be an algebraic curve that degenerates into n components C1, . . . , Cn. ◮ Let G = (V , E), where vertex vi corresponds to the component Ci and |Ci ∩ Cj| = # edges from vi to vj.

Motivation: Arithmetic Geometry (Lorenzini ’89)

◮ Let C be an algebraic curve that degenerates into n components C1, . . . , Cn. ◮ Let G = (V , E), where vertex vi corresponds to the component Ci and |Ci ∩ Cj| = # edges from vi to vj. ◮ Let d be the vector of self-intersection numbers.

Motivation: Arithmetic Geometry (Lorenzini ’89)

◮ Let C be an algebraic curve that degenerates into n components C1, . . . , Cn. ◮ Let G = (V , E), where vertex vi corresponds to the component Ci and |Ci ∩ Cj| = # edges from vi to vj. ◮ Let d be the vector of self-intersection numbers. ◮ The critical group K(G, d, r) ∼ = group of components of the N´ eron model of the Jacobian of C.

Arithmetical Structures on Paths

Proposition

In Pn, D is the only d-structure with di ≥ 2 for 1 < i < n.

Arithmetical Structures on Paths

Proposition

In Pn, D is the only d-structure with di ≥ 2 for 1 < i < n.

Theorem

r = (r1, . . . , rn) ∈ Zn

>0 is an r-structure on Pn if and only if

(i) r is primitive and r1 = rn = 1, (ii) ri | (ri−1 + ri+1) for all i = 2, . . . , n − 1.

Arithmetical Structures on Paths

Proposition

In Pn, D is the only d-structure with di ≥ 2 for 1 < i < n.

Theorem

r = (r1, . . . , rn) ∈ Zn

>0 is an r-structure on Pn if and only if

(i) r is primitive and r1 = rn = 1, (ii) ri | (ri−1 + ri+1) for all i = 2, . . . , n − 1. D = (1, 1), r = (1, 1) (Laplacian a. s.).

Arithmetical Structures on Paths

Proposition

In Pn, D is the only d-structure with di ≥ 2 for 1 < i < n.

Theorem

r = (r1, . . . , rn) ∈ Zn

>0 is an r-structure on Pn if and only if

(i) r is primitive and r1 = rn = 1, (ii) ri | (ri−1 + ri+1) for all i = 2, . . . , n − 1. D = (1, 1), r = (1, 1) (Laplacian a. s.). D = (1, 2, 1), r = (1, 1, 1) d = (2, 1, 2), r = (1, 2, 1)   2 −1 −1 1 −1 −1 2     1 2 1   = 0

Arithmetical Structures on Paths

Theorem

r = (r1, . . . , rn) ∈ Zn

>0 is an r-structure on Pn if and only if

(i) r is primitive and r1 = rn = 1, (ii) ri | (ri−1 + ri+1) for all i = 2, . . . , n − 1. D = (1, 1), r = (1, 1) (Laplacian a. s.). D = (1, 2, 1), r = (1, 1, 1) d = (2, 1, 2), r = (1, 2, 1) D = (1, 2, 2, 1), r = (1, 1, 1, 1) d = (2, 1, 3, 1), r = (1, 2, 1, 1) d = (1, 3, 1, 2), r = (1, 1, 2, 1) d = (3, 1, 2, 2), r = (1, 3, 2, 1) d = (2, 2, 1, 3), r = (1, 2, 3, 1)

Arithmetical Structures on Cycles

Proposition

In Cn, D = ¯ 2 is the only d-structure with di ≥ 2 for all i.

Arithmetical Structures on Cycles

Proposition

In Cn, D = ¯ 2 is the only d-structure with di ≥ 2 for all i.

Theorem

Let r = (r1, . . . , rn) ∈ Zn

>0 be primitive. Then r is an r-structure

• n Cn if and only if

ri | (ri−1 + ri+1) for all i, with the indices taken modulo n.

Arithmetical Structures on Cycles

Proposition

In Cn, D = ¯ 2 is the only d-structure with di ≥ 2 for all i.

Theorem

Let r = (r1, . . . , rn) ∈ Zn

>0 be primitive. Then r is an r-structure

• n Cn if and only if

ri | (ri−1 + ri+1) for all i, with the indices taken modulo n. D = (2, 2), r = (1, 1) d = (1, 4), r = (2, 1) d = (4, 1), r = (1, 2)

Arithmetical Structures on Cycles

Theorem

Let r = (r1, . . . , rn) ∈ Zn

>0 be primitive. Then r is an r-structure

• n Cn if and only if

ri | (ri−1 + ri+1) for all i, with the indices taken modulo n. D = (2, 2), r = (1, 1) d = (1, 4), r = (2, 1) d = (4, 1), r = (1, 2) D = (2, 2, 2), r = (1, 1, 1) d = (3, 1, 3), r = (1, 2, 1) (3 permutations) d = (2, 1, 5), r = (2, 3, 1) (6 permutations)

Counting Arithmetical Structures

Theorem (Ben Braun, Hugo Corrales, Scott Corry, , Darren Glass, Nathan Kaplan, Jeremy Martin, Gregg Musiker Carlos Valencia, 2018)

Let Pn and Cn denote the path graph and cycle graph on n vertices, respectively. Then

| Arith(Pn)| = Cn−1 = 1 n 2n − 2 n − 1

• ,

| Arith(Cn)| = (2n−1)Cn−1 = 2n − 1 n − 1

• .

K(Pn, d, r) = 0, K(Cn, d, r) = Zr(1), r(1) = #1’s in r.

Arithmetical Structures on Paths

Proposition

Given a triangulation T of an (n + 1)-gon, define d(T) = (d0, d1, . . . , dn) by di = # triangles incident to vertex i. The map d gives a bijection between triangulations of an (n + 1)-gon and arithmetical d-structures on Pn.

Arithmetical Structures on Paths

Proposition

Given a triangulation T of an (n + 1)-gon, define d(T) = (d0, d1, . . . , dn) by di = # triangles incident to vertex i. The map d gives a bijection between triangulations of an (n + 1)-gon and arithmetical d-structures on Pn. Conway-Coxeter Frieze Patterns

1 1 1 1 1 1 1 1 1 1 1 1 1 1 · · · 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 · · · 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Arithmetical Structures on Paths

Proposition

Given a triangulation T of an (n + 1)-gon, define d(T) = (d0, d1, . . . , dn) by di = # triangles incident to vertex i. The map d gives a bijection between triangulations of an (n + 1)-gon and arithmetical d-structures on Pn. Conway-Coxeter Frieze Patterns

Arithmetical Structures on Paths

Proposition

Given a triangulation T of an (n + 1)-gon, define d(T) = (d0, d1, . . . , dn) by di = # triangles incident to vertex i. The map d gives a bijection between triangulations of an (n + 1)-gon and arithmetical d-structures on Pn. Conway-Coxeter Frieze Patterns

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 · · · 2 2 1 4 2 1 4 1 3 2 1 4 2 1 4 1 3 2 · · · 5 1 3 7 1 3 3 2 5 1 3 7 1 3 3 2 5 · · · 3 2 2 5 3 2 2 5 3 2 2 5 3 2 2 5 3 2 · · · 1 3 3 2 5 1 3 7 1 3 3 2 5 1 3 7 1 · · · 2 1 4 1 3 2 1 4 2 1 4 1 3 2 1 4 2 1 · · · 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Arithmetical Structures on Stars

Let Kn,1 denote the star graph with n leaves.

Theorem

The d-arithmetical structures on Kn,1 are the positive integer solutions to d0 =

n

• i=1

1 di . Each such solution is an Egyptian fraction representation of d0. Observation: There is no closed form for the sequence | Arith(Kn,1)|. 1, 2, 14, 263, 13462, 2104021, . . .

Arithmetical Structures on Dynkin Graphs

x y v0 v1 · · · v¸

Theorem (Kassie Archer, Abigail Bishop, Alexander Diaz-Lopez, , Darren Glass, Joel Lowsma (2018))

Let Dn denote the Dynkin graph with n vertices. Then 14Cn−3 ≤ | Arith(Dn)| ≤ Cn−3 3 (4n4 − 20n3 + 38n2 − 28n − 36), where Cn is the nth Catalan number. Moreover, |K(Dn; d, r)| is a cyclic group of order |K(Dn; d, r)| = r0 rxryrℓ .