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 0

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 v j at every boundary node j 4 3 1 0 0 2 1

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? 4 3 0 2 1

The Dirichlet solution ◮ Every wire ij ∈ E has a (real or complex) conductance c ij ◮ Voltages and conductances satisfy � c ij ( v i − v j ) = 0 j ∼ i 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 Interior voltages solved! conductances labeled: 55 71 1 4 1 4 3 3 1 0 1 0 2 5 2 5 52 71 Let i ∈ V ◦ be the top vertex. Can check: � 55 � � 55 � � 55 � � 71 − 52 c ij ( v i − v j ) = 1 71 − 1 +3 +4 71 − 0 = 0 � 71 j ∼ i

Energies Definition The energy dissipated by an edge ij ∈ E is e ij = c ij ( v i − v j ) 2 Example Let ∆ = ( c 1 + c 3 + c 4 )( c 2 + c 3 + c 5 ) − c 2 3 . We have     1 c 1 c 2 c 1 ( c 3 c 5 + c 4 ( c 2 + c 3 + c 5 )) 2 e 1     c 2 ( c 3 c 4 + ( c 1 + c 3 + c 4 ) c 5 ) 2 c 3 e 2     = 1     c 3 ( c 2 c 4 − c 1 c 5 ) 2 e 3     ∆ 2     c 4 ( c 2 c 3 + c 1 ( c 2 + c 3 + c 5 )) 2 e 4 c 4 c 5 c 5 ( c 1 c 3 + c 2 ( c 1 + c 3 + c 4 )) 2 e 5 0

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 ∆ = ( c 1 + c 3 + c 4 )( c 2 + c 3 + c 5 ) − c 2 3 . We have     1 c 1 c 2 c 1 ( c 3 c 5 + c 4 ( c 2 + c 3 + c 5 )) 2 e 1     c 2 ( c 3 c 4 + ( c 1 + c 3 + c 4 ) c 5 ) 2 c 3 e 2     = 1     c 3 ( c 2 c 4 − c 1 c 5 ) 2 e 3     ∆ 2     c 4 ( c 2 c 3 + c 1 ( c 2 + c 3 + c 5 )) 2 e 4 c 4 c 5 c 5 ( c 1 c 3 + c 2 ( c 1 + c 3 + c 4 )) 2 e 5 0

e -harmonic functions Definition Fix a graph G , boundary voltages v , and (real or complex) edge energies e . A function h ∈ C V 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: a b 25 a − 5 25 b − 5 25 b − 5 25 a − 5 5 5 1 0 1 0 25 a − 5 25 a − 5 25 b − 5 25 b − 5 a b √ √ where a = 1 5) and b = 1 10 (5 − 10 (5 + 5). Can check energy of middle edge ij : � 1 � 2 √ √ 5) − 1 e ij = c ij ( v i − v j ) 2 = 5 10(5 + 10(5 − 5) = 1 �

Section II: Dirichlet Arrangements

Arrangements ◮ An arrangement is a set of affine hyperplanes (in R n or C n ) ◮ Arrangements in R n divide the space into chambers ◮ E.g. — Arrangements in R 2 and R 3 :

Dirichlet arrangements ◮ Recall: G a graph with boundary voltages v Definition The Dirichlet arrangement A G , v consists of two types of hyperplanes, corresponding to the edges of G , with coordinates indexed by interior nodes: ◮ A hyperplane x i = v j for every edge ij with j ∈ ∂ V ◮ A hyperplane x i = x j for every edge ij not meeting ∂ V r x r = x s x s = 1 1 0 x s = 0 s x r = 0 x r = 1

Master functions ◮ Let A be an arrangement of k hyperplanes in C n , defined by affine functions f 1 , . . . , f k : C n → C ◮ Master function of A with weights b ∈ C k is multivalued C n → C given by k � f i ( x ) b i Φ b ( x ) = i =1 ◮ A critical point x ∈ C n of Φ b : k � ∂ ∂ f i b i log Φ b ( x ) = f i ( x ) = 0 ∂ x j ∂ x j i =1 for all j = 1 , . . . , n

Example ◮ Arrangement A in C 2 defined by A : x = 0 , y = 0 , x + y − 1 = 0 ◮ Master function with weights b = ( b 1 , b 2 , b 3 ) ∈ C 3 : Φ b ( x , y ) = x b 1 y b 2 ( x + y − 1) b 3 ◮ Critical point equations: � b 1 � x + y − 1 , b 2 b 3 b 3 ∇ log Φ b ( x , y ) = x + y + = (0 , 0) x + y − 1 � � b 1 b 2 ( x , y ) = , b 1 + b 2 + b 3 b 1 + b 2 + b 3

Why Dirichlet arrangements? Theorem Fix edge energies e ∈ C E . The e -harmonic functions on ( G , v ) are the critical points of the master function of A G , v with weights e . Important example When ( G , ∂ V ) is complete , A G , 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 A G , 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 A G , v , and no others Example √ √ Let a = 1 5) and b = 1 10 (5 − 10 (5 + 5) with all energies 1 a or b ( b , a ) 1 0 ( a , b ) b or 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: a b 1 0 1 0 a b √ √ where a = 1 5) and b = 1 10 (5 − 10 (5 + 5).

Counting bounded chambers ◮ Chromatic polynomial χ G 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 # current flows β ( � G ) = # e -harmonic functions # bounded chambers ( | ∂ V | − 2)! ◮ This formula also has applications to a visibility problem

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

Visibility sets Sets of top-dimensional faces of P Chambers of vis( P ) � visible from different points in R n

Order polytopes Definition Let P be a finite poset. The order polytope O ( P ) is the convex polytope in R P of all order-preserving functions P → [0 , 1] Example If every pair in P = { x 1 , . . . , x n } is incomparable, then O ( P ) = [0 , 1] n is the unit hypercube in R n Example If P = { x 1 , . . . , x n } is totally ordered, then O ( P ) is an n -simplex in R n

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 A G , v . 1 Hasse diagram of P 0 � G ( G , v )

Counting visibility sets Theorem (L.) The number of visibility sets of the order polytope O ( P ) is 1 2 α ( � 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 = { x 1 , . . . , x n } totally ordered ◮ O ( P ) is an n -simplex in R n G ) = 2 n +1 − 1 visibility sets (all but full set of faces) 2 α ( � 1 ◮ ◮ All but β ( � G ) = 1 visible from far away (the empty set) 1 x n . . . . . . . . . x 2 x 1 0 Hasse diagram of P � G ( G , v )

Section III: The Response Matrix  c 1 ( S − c 1 ) − c 1 c 2 − c 1 c n  · · · − c 1 c 2 c 2 ( S − c 2 ) − c 2 c n Λ = 1 · · ·    . . .  ... . . . S   . . .   − c 1 c n − c 2 c n c n ( S − c n ) · · ·

Electrical response ◮ G a graph with boundary ∂ V and conductances c ◮ We know current across interior node i is 0: � c ij ( v i − v j ) = 0 j ∼ i ◮ 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 c 1 c 4 � 1 � c 5 − 1 Λ = R · − 1 1 c 2 c 3 R = c 1 c 2 c 3 + c 1 c 2 c 4 + c 1 c 2 c 5 + c 1 c 3 c 4 + c 1 c 3 c 5 + c 2 c 3 c 4 + c 3 c 4 c 5 c 1 c 3 + c 1 c 4 + c 1 c 5 + c 2 c 3 + c 2 c 4 + c 2 c 5 + c 3 c 5 + c 4 c 5

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