Counting Arithmetical Structures

Transcription

1 Counting Arithmetical Structures Luis David García 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

2 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.

3 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 with M 6...

4 Gutenberg-Richter Law (in terms of Energy) Shallow worldwide earthquakes (Global Centroid Moment Tensor Project)

5 Gutenberg-Richter Law (in terms of Energy) 1. The Size of Earthquakes 11 Shallow worldwide earthquakes (Global Centroid Shallow Moment worldwide Tensor earthquakes Project) (centroid moment tensor catalog): after Kagan D(E) / 1/E ) ln D(E) = constant ln E Geophys J Int 2002 Tectonophys 2010

6 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?

7 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.

8 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 Model Structure and Definitions 3 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, Tang and Wiesenfeld model of grain. self-orgainized critical- This process continues until all sites are stable.

9 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, Tang and Wiesenfeld model of Markov self-orgainized process. criticaly mean-squared velocity difference scales with distance, Mathematical Model for Sandpiles In 1987, Bak, Tang, and Wiesenfeld proposed the following model that captures important features of self-organized criticality. Model Structure and Definitions 3 Start this process on an empty grid. At first there is little activity.

10 Distribution of sandpiles Distribution of avalanche sizes in a grid.

11 Abelian Sandpile Model on Graphs [Dhar (1990)] v 1 v 2 Let Γ = (V, E, s) denote a finite, connected, loopless multigraph with a distinguished vertex s called the sink. s v 4 v 3 A configuration over Γ is a function σ : V \ {s} N. 2 0 s 4 1

12 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.

13 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}.

14 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

15 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 s s

16 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 s s s

17 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.

18 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.

19 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 4 s v 2 v L =

20 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 k1 Z k2 Z kn.

21 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 k1 Z k2 Z kn. v 1 v 4 s v 2 v L =

22 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 k1 Z k2 Z kn. v 1 s v L = v 4 v 3

23 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 k1 Z k2 Z kn. v 1 v 2 s S(Γ) = Z 8 Z 24, S(Γ) = 8 24 = 192. v 4 v 3

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(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 k1 Z k2 Z kn. 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).

25 Sandpiles on a grid Model Structure and D Figure 1.2: The Bak, Tang and Wiesenfeld model of self-orgainized criticality. 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. observed in the way mean-squared velocity difference scales with distance or in the The spatial sandpile fractal group structure of theof4 4-grid regions of is Zhigh 2 8 dissipation. Z 1320 Z Systems that exhibit this significant correlation with power-law decay are said to have critical correlations. For equilibrium systems, such as a

26 Sandpiles on a grid Model Structure and D Figure 1.2: The Bak, Tang and Wiesenfeld model of self-orgainized criticality. 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. observed in the way mean-squared velocity difference scales with distance or in the The spatial sandpile fractal group structure of theof4 4-grid regions of is Zhigh 2 8 dissipation. Z 1320 Z Systems that exhibit this significant correlation with power-law decay are said to have critical correlations. For equilibrium systems, such as a

27 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 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.

28 Identity sandpile in the square grid Color scheme: black=0, yellow=1, blue=2, and red=3.

29 Arithmetical Structures Let G be a finite, simple, connected graph with n 2 vertices.

30 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.)

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

32 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 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.

33 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.

34 Arithmetical Structures Arith(G) := {(d, r) (d, r) is an arithmetical structure on G}.

35 Arithmetical Structures Arith(G) := {(d, r) (d, r) is an arithmetical structure on G}. (G, d, r) is called an arithmetical graph.

36 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) = Z n / im L(G, d) = Z K(G, d, r).

37 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) = Z n / 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).

38 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) = Z n / 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.

39 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) = Z n / 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.)

40 Motivation: Arithmetic Geometry (Lorenzini 89) Let C be an algebraic curve that degenerates into n components C 1,..., C n.

41 Motivation: Arithmetic Geometry (Lorenzini 89) Let C be an algebraic curve that degenerates into n components C 1,..., C n. Let G = (V, E), where vertex v i corresponds to the component C i and C i C j = # edges from v i to v j.

42 Motivation: Arithmetic Geometry (Lorenzini 89) Let C be an algebraic curve that degenerates into n components C 1,..., C n. Let G = (V, E), where vertex v i corresponds to the component C i and C i C j = # edges from v i to v j. Let d be the vector of self-intersection numbers.

43 Motivation: Arithmetic Geometry (Lorenzini 89) Let C be an algebraic curve that degenerates into n components C 1,..., C n. Let G = (V, E), where vertex v i corresponds to the component C i and C i C j = # edges from v i to v j. Let d be the vector of self-intersection numbers. The critical group K(G, d, r) = group of components of the Néron model of the Jacobian of C.

44 Arithmetical Structures on Paths Proposition In P n, D is the only d-structure with d i 2 for 1 < i < n.

45 Arithmetical Structures on Paths Proposition In P n, D is the only d-structure with d i 2 for 1 < i < n. Theorem r = (r 1,..., r n ) Z n >0 is an r-structure on P n if and only if (i) r is primitive and r 1 = r n = 1, (ii) r i (r i 1 + r i+1 ) for all i = 2,..., n 1.

46 Arithmetical Structures on Paths Proposition In P n, D is the only d-structure with d i 2 for 1 < i < n. Theorem r = (r 1,..., r n ) Z n >0 is an r-structure on P n if and only if (i) r is primitive and r 1 = r n = 1, (ii) r i (r i 1 + r i+1 ) for all i = 2,..., n 1. D = (1, 1), r = (1, 1) (Laplacian a. s.).

47 Arithmetical Structures on Paths Proposition In P n, D is the only d-structure with d i 2 for 1 < i < n. Theorem r = (r 1,..., r n ) Z n >0 is an r-structure on P n if and only if (i) r is primitive and r 1 = r n = 1, (ii) r i (r i 1 + r i+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) =

48 Arithmetical Structures on Paths Theorem r = (r 1,..., r n ) Z n >0 is an r-structure on P n if and only if (i) r is primitive and r 1 = r n = 1, (ii) r i (r i 1 + r i+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)

49 Arithmetical Structures on Cycles Proposition In C n, D = 2 is the only d-structure with d i 2 for all i.

50 Arithmetical Structures on Cycles Proposition In C n, D = 2 is the only d-structure with d i 2 for all i. Theorem Let r = (r 1,..., r n ) Z n >0 on C n if and only if be primitive. Then r is an r-structure r i (r i 1 + r i+1 ) for all i, with the indices taken modulo n.

51 Arithmetical Structures on Cycles Proposition In C n, D = 2 is the only d-structure with d i 2 for all i. Theorem Let r = (r 1,..., r n ) Z n >0 on C n if and only if be primitive. Then r is an r-structure r i (r i 1 + r i+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)

52 Arithmetical Structures on Cycles Theorem Let r = (r 1,..., r n ) Z n >0 on C n if and only if be primitive. Then r is an r-structure r i (r i 1 + r i+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) d = (2, 1, 5), r = (2, 3, 1) (3 permutations) (6 permutations)

53 Counting Arithmetical Structures Theorem (Ben Braun, Hugo Corrales, Scott Corry,, Darren Glass, Nathan Kaplan, Jeremy Martin, Gregg Musiker Carlos Valencia, 2018) Let P n and C n denote the path graph and cycle graph on n vertices, respectively. Then Arith(P n ) = C n 1 = 1 n ( ) 2n 2, Arith(C n ) = (2n 1)C n 1 = n 1 ( ) 2n 1. n 1 K(P n, d, r) = 0, K(C n, d, r) = Z r(1), r(1) = #1 s in r.

54 Arithmetical Structures on Paths Proposition Given a triangulation T of an (n + 1)-gon, define d(t ) = (d 0, d 1,..., d n ) by d i = # triangles incident to vertex i. The map d gives a bijection between triangulations of an (n + 1)-gon and arithmetical d-structures on P n.

55 Arithmetical Structures on Paths Proposition Figure Given 2: ajohn triangulation Horton Conway T of an (left) (n(born + 1)-gon, 1937) define and Harold Scott MacDonald Coxeter (right) ( ). d(t ) = (d 0, d 1,..., d n ) by d i = # triangles incident to vertex i. Harold The Scott map dmacdonald gives a bijection Coxeter between (1907 triangulations 2003) (see Figure of an 2), who introduced (n + and 1)-gon studied andthem arithmetical in the early d-structures 1970s [5]. on Before P n. we are going to define frieze patterns properly, you might want to have a look at some first examples in Figure 3 (the green [italic] and red [bold] colours will be explained later). Conway-Coxeter Frieze Patterns (a) A simple frieze pattern.

56 Arithmetical Structures on Paths Proposition Given a triangulation T of an (n + 1)-gon, define d(t ) = (d 0, d 1,..., d n ) by d i = # triangles incident to vertex i. The map d gives a bijection between triangulations of an (n + 1)-gon and arithmetical d-structures on P n. Conway-Coxeter Frieze Patterns

57 Figure 2: John Horton Conway (left) (born 1937) and Harold Scott MacDonald Arithmetical Coxeter Structures (right) (1907 on 2003). Paths Proposition Given a triangulation T of an (n + 1)-gon, define Harold Scott MacDonald Coxeter ( ) (see Figure 2), who introduced and studied them in the early 1970s [5]. Before we are going to define frieze patterns properly, you might want to have a look at some first examples in Figure 3 (the green [italic] and red [bold] colours will be explained later). d(t ) = (d 0, d 1,..., d n ) by d i = # triangles incident to vertex i The map 2 d1gives 2 1a bijection between triangulations of an (n + 1)-gon 1 1and1 arithmetical d-structures on1 P1 n. 1 (a) A simple frieze pattern. Conway-Coxeter Frieze Patterns (b) A more complicated frieze pattern.

58 Arithmetical Structures on Stars Let K n,1 denote the star graph with n leaves. Theorem The d-arithmetical structures on K n,1 are the positive integer solutions to n 1 d 0 =. d i Each such solution is an Egyptian fraction representation of d 0. i=1 Observation: There is no closed form for the sequence Arith(K n,1 ). 1, 2, 14, 263, 13462, ,...

59 Kassie Archer, Abigail C. Bishop, Alexander Diaz Luis D. García Puente, Darren Glass, Joel Louw Arithmetical Structures on Dynkin Graphs rx x v0 v1 r0 v r ry y Theorem (Kassie Archer, Abigail Bishop, Alexander 1 Introduction, Darren Glass, Joel Lowsma (2018)) Diaz-Lopez, arithmetical structure on an graph is typically Let Dn denoteanthe Dynkin graph with vertices. Then given by a pair of vect 14Cn 3 We note that the existence of a unique vector r whose entries have equal to one follows Cn 3from 4the fact 3that the 2nullspace of A D is one choice ofn ) r is literature, it more con Arith(D standard (4nin the 20n + 38nwe often 28n find 36), the vector scaled 3by a factor of rxr0ry. It is straightforward to check r 0 = r0 rxr0ry = rx rxr0ry ry rrx0 ry = r x ry. In other words, the structure takes where Cn is the nth Catalan number. Moreover, K(Dn ; d, r) is a cyclic group of order a K(Dn ; d, r) = r0. rx ry r` ab r 1 r