Electrical Networks and Hyperplane Arrangements Bob Lutz - - PowerPoint PPT Presentation

Electrical Networks and Hyperplane Arrangements

Bob Lutz

Mathematical Sciences Research Institute

June 7, 2019

Partially supported by NSF grants DMS-1401224 and DMS-1701576

Section I: Electrical Networks

1

What is an electrical network?

◮ A connected graph G = (V , E) (edges = wires) ◮ A set ∂V ⊆ V of at least 2 boundary nodes ◮ A (real or complex) voltage vj at every boundary node j

1 1 2 3 4

The Dirichlet problem

◮ Electrical current flows from higher voltages to lower voltages ◮ Consider the interior V ◦ = V \ ∂V ◮ What are the voltages at the interior nodes?

1 2 3 4

The Dirichlet solution

◮ Every wire ij ∈ E has a (real or complex) conductance cij ◮ Voltages and conductances satisfy

• j∼i

cij(vi − vj) = 0 at every interior node i ∈ V ◦ ◮ “The current across every interior node is 0” ◮ Uniquely determines the interior voltages (for generic c)

Example

Network with conductances labeled: Interior voltages solved! 1 4 2 5 3

1

55 71 52 71

1 4 2 5 3

1

Let i ∈ V ◦ be the top vertex. Can check:

• j∼i

cij(vi−vj) = 1 55 71 − 1

• +3

55 71 − 52 71

• +4

55 71 − 0

• = 0

Energies

Definition

The energy dissipated by an edge ij ∈ E is eij = cij(vi − vj)2

Example

Let ∆ = (c1 + c3 + c4)(c2 + c3 + c5) − c2

• 3. We have

c1 c4 c2 c5 c3

1

      e1 e2 e3 e4 e5       = 1 ∆2       c1(c3c5 + c4(c2 + c3 + c5))2 c2(c3c4 + (c1 + c3 + c4)c5)2 c3(c2c4 − c1c5)2 c4(c2c3 + c1(c2 + c3 + c5))2 c5(c1c3 + c2(c1 + c3 + c4))2      

Question

◮ The map c → e is rational (polynomial/polynomial) ◮ What do the fibers look like? ◮ Equivalently, which conductances and interior voltages produce the energies e?

Example

Let ∆ = (c1 + c3 + c4)(c2 + c3 + c5) − c2

• 3. We have

c1 c4 c2 c5 c3

1

      e1 e2 e3 e4 e5       = 1 ∆2       c1(c3c5 + c4(c2 + c3 + c5))2 c2(c3c4 + (c1 + c3 + c4)c5)2 c3(c2c4 − c1c5)2 c4(c2c3 + c1(c2 + c3 + c5))2 c5(c1c3 + c2(c1 + c3 + c4))2      

e-harmonic functions

Definition

Fix a graph G, boundary voltages v, and (real or complex) edge energies e. A function h ∈ CV is e-harmonic on (G, v) if

• 1. There are conductances c with respect to which h is the

voltage function for the network

• 2. The energies dissipated are e.

Interesting Problem

Describe the set of e-harmonic functions for a given e.

Example

Let all edge energies e be 1. Fix boundary voltages 0 and 1. There are two e-harmonic functions, with conductances labeled: 25b − 5 25a − 5 25a − 5 25b − 5 5

1 b a

25a − 5 25b − 5 25b − 5 25a − 5 5

1 a b

where a = 1

10(5 −

√ 5) and b = 1

10(5 +

√ 5). Can check energy of middle edge ij: eij = cij(vi − vj)2 = 5 1 10(5 + √ 5) − 1 10(5 − √ 5) 2 = 1

Section II: Dirichlet Arrangements

Arrangements

◮ An arrangement is a set of affine hyperplanes (in Rn or Cn) ◮ Arrangements in Rn divide the space into chambers ◮ E.g. — Arrangements in R2 and R3:

Dirichlet arrangements

◮ Recall: G a graph with boundary voltages v

Definition

The Dirichlet arrangement AG,v consists of two types of hyperplanes, corresponding to the edges of G, with coordinates indexed by interior nodes: ◮ A hyperplane xi = vj for every edge ij with j ∈ ∂V ◮ A hyperplane xi = xj for every edge ij not meeting ∂V r s

1

xr = 0 xr = 1 xs = 0 xs = 1 xr = xs

Master functions

◮ Let A be an arrangement of k hyperplanes in Cn, defined by affine functions f1, . . . , fk : Cn → C ◮ Master function of A with weights b ∈ Ck is multivalued Cn → C given by Φb(x) =

k

• i=1

fi(x)bi ◮ A critical point x ∈ Cn of Φb: ∂ ∂xj log Φb(x) =

k

• i=1

∂fi ∂xj bi fi(x) = 0 for all j = 1, . . . , n

Example

◮ Arrangement A in C2 defined by A : x = 0, y = 0, x + y − 1 = 0 ◮ Master function with weights b = (b1, b2, b3) ∈ C3: Φb(x, y) = xb1yb2(x + y − 1)b3 ◮ Critical point equations: ∇ log Φb(x, y) = b1 x + b3 x + y − 1 , b2 y + b3 x + y − 1

• = (0, 0)

(x, y) =

• b1

b1 + b2 + b3 , b2 b1 + b2 + b3

Why Dirichlet arrangements?

Theorem

Fix edge energies e ∈ CE. The e-harmonic functions on (G, v) are the critical points of the master function of AG,v with weights e.

Important example

When (G, ∂V ) is complete, AG,v is a discriminantal arrangement:

So what?

Critical points of master functions are well studied! ◮ Interior point methods: logarithmic barrier functions and analytic centers . . . especially for discriminantal arrangements. ◮ Real algebraic geometry: solution of B. and M. Shapiro conjecture on Wronskians ◮ Quantum integrable systems: Bethe ansatz in the Gaudin model

◮ Main problem: diagonalize a set of commuting linear operators (Hamiltonians) ◮ Critical points of master functions parameterize the eigenvectors

Consequences

Corollary

Suppose the boundary voltages are real, so AG,v lives in real space. If the energies e are all positive, then ◮ The e-harmonic functions take all real values ◮ There is exactly one e-harmonic function in each bounded chamber of AG,v, and no others

Example

Let a = 1

10(5 −

√ 5) and b = 1

10(5 +

√ 5) with all energies 1

1 a or b b or a (a, b) (b, a)

Current flow

◮ Real-valued e-harmonic functions induce current flows ◮ Edges directed from higher voltages to lower ◮ # nonzero current flows = # e-harmonic functions = # bounded chambers

Example

In the running example:

1 b a 1 a b

where a = 1

10(5 −

√ 5) and b = 1

10(5 +

√ 5).

Counting bounded chambers

◮ Chromatic polynomial χG counts proper vertex colorings of G ◮ The beta invariant is β(G) = |χ′

G(1)|

◮ β(G) = 0 iff G is disconnected by removing single vertex

Theorem (L.)

Obtain a graph G by adding edges between all boundary vertices. Then for generic energies e (including all positive) we have β( G) (|∂V | − 2)! = # current flows # e-harmonic functions # bounded chambers ◮ This formula also has applications to a visibility problem

Visibility arrangements

Definition

Let P be a convex n-polytope in Rn. The visibility arrangement vis(P) is the set of affine spans of the top-dimensional faces of P

Visibility sets

Chambers of vis(P) Sets of top-dimensional faces of P visible from different points in Rn

Order polytopes

Definition

Let P be a finite poset. The order polytope O(P) is the convex polytope in RP of all order-preserving functions P → [0, 1]

Example

If every pair in P = {x1, . . . , xn} is incomparable, then O(P) = [0, 1]n is the unit hypercube in Rn

Example

If P = {x1, . . . , xn} is totally ordered, then O(P) is an n-simplex in Rn

Visibility arrangements of order polytopes

Proposition (Stanley)

Let P be a finite poset. The visibility arrangement vis(O(P)) of the order polytope of P is a Dirichlet arrangement AG,v. Hasse diagram of P

1

(G, v)

• G

Counting visibility sets

Theorem (L.)

The number of visibility sets of the order polytope O(P) is 1 2α( G), where α( G) is the number of acyclic orientations of

• G. Of these

visibility sets, all but β( G) are visible from far away, where β( G) is the beta invariant.

Example

◮ P = {x1, . . . , xn} totally ordered ◮ O(P) is an n-simplex in Rn ◮

1 2α(

G) = 2n+1 − 1 visibility sets (all but full set of faces) ◮ All but β( G) = 1 visible from far away (the empty set) x1 x2 xn

. . .

Hasse diagram of P

. . .

1

(G, v)

. . .

• G

Section III: The Response Matrix

Λ = 1 S      c1(S − c1) −c1c2 · · · −c1cn −c1c2 c2(S − c2) · · · −c2cn . . . . . . ... . . . −c1cn −c2cn · · · cn(S − cn)     

Electrical response

◮ G a graph with boundary ∂V and conductances c ◮ We know current across interior node i is 0:

• j∼i

cij(vi − vj) = 0 ◮ Current across boundary node not necessarily 0 ◮ Map from boundary voltages v to boundary currents is linear C∂V → C∂V

Definition

The response matrix is the matrix Λ = Λ(G, ∂V , c) of the map from boundary voltages to boundary currents.

Entries of response matrix

◮ Response matrix Λ maps boundary voltages to boundary currents ◮ Entries of Λ are rational functions in the conductances c1 c2 c4 c3 c5 Λ = R · 1 −1 −1 1

• R = c1c2c3 + c1c2c4 + c1c2c5 + c1c3c4 + c1c3c5 + c2c3c4 + c3c4c5

c1c3 + c1c4 + c1c5 + c2c3 + c2c4 + c2c5 + c3c5 + c4c5

Groves

◮ We are interested in the trace of Λ

Definition

A grove is a forest F ⊆ E in G such that ◮ F meets every interior node ◮ Every tree in F meets ∂V at least once

Trace formula

◮ Let F0 (resp., F1) be the set of groves containing no (resp., exactly 1) path between boundary nodes A grove in F1 Two groves in F0

Proposition

The trace of the response matrix can be written as 1 2 tr Λ =

• F∈F1
• ij∈F cij
• F∈F0
• ij∈F cij

.

Trace formula

Proposition

The trace of the response matrix can be written as 1 2 tr Λ =

• F∈F1
• ij∈F cij
• F∈F0
• ij∈F cij

. c2 c1 c3 c4 c5 1 2 tr Λ = c1c2c3 + c1c2c4 + c1c2c5 + c1c3c4 + c1c3c5 + c2c3c4 + c3c4c5 c1c3 + c1c4 + c1c5 + c2c3 + c2c4 + c2c5 + c3c5 + c4c5

Interlacing result

Theorem (L.)

Let x, y ∈ RE with all entries of y positive. The zeros and poles of tr Λ interlace along the line x + ty. Two sequences of points on a line interlace along the line if they alternate like so: · · · · · ·

Example

Consider the star network with n boundary nodes and conductances a = (c1, . . . , cn):

1 2 tr Λ =

• i<j

cicj

• i

ci

◮ Consider the line (−1, . . . , −1, n − 1) + t(1, . . . , 1) in Rn ◮ The zeros are at t = ±1 and the sole pole is at t = 0 ◮ Case n = 2 illustrated on the right