planar graph embeddings and stat mech Richard Kenyon (Brown - PowerPoint PPT Presentation

planar graph embeddings and stat mech Richard Kenyon (Brown University) Wednesday, May 11, 16

In 2D stat mech models, appropriate graph embeddings are important e.g. Bond percolation on Z 2 . p c = 1 2 What about unequal probabilities? q q q q q q q q q q θ q q critical if: q q q p 3 + 3 p 2 q − 3 p 2 − 3 pq + 1 = 0 θ = θ ( p, q ) Wednesday, May 11, 16

In 2D stat mech models, appropriate graph embeddings are important Random walks/spanning trees harmonic embedding, BSST, LSW, square tiling Georgakopoulos circle packing Angel, Barlow, Gurel-Gurevich, Nachmias trapezoid tiling Hutchcroft, Peres (Kenyon, She ffi eld) Dimer models T-graphs (Kenyon, Mercat, Smirnov) Ising model K-graphs FK (random cluster) model isoradial graphs (Abrams, Kenyon) area-1 rectangulations bipolar orientations Schnyder embedding (Schnyder, , X. Sun, Watson) Schnyder woods Random planar maps conformal (KPZ, Duplantier, Miller, She ffi eld) Wednesday, May 11, 16

1. T-graphs and dimers 2. Convex embeddings of a planar graph 3. Harmonic embeddings 4. Discrete analytic functions 5. Fixed-area rectangulations Wednesday, May 11, 16

Smith diagram of a planar network [BSST 1939] (with a harmonic function) v 1 3 1 1. 2 2. 4. 6 4 3. 5. 9 5 8 7 7. 6. 8. 10 10. 9. 12 11 11. 12. v 0 vertex = horizontal line voltage = y -coordinate edge = rectangle current = width (width/height) conductance = aspect ratio energy = area Wednesday, May 11, 16

Smith diagram of a planar network [BSST 1939] (with a harmonic function) v 1 3 1 1. 2 3 1 2 2. 4. 6 4 4 6 3. 5. 9 5 9 5 8 7 7. 8 7 6. 8. 10 10 10. 9. 12 11 11. 12 11 12. v 0 vertex = horizontal line voltage = y -coordinate edge = rectangle current = width (width/height) conductance = aspect ratio energy = area Wednesday, May 11, 16

Thm(Dehn 1903): An a × b rectangle can be tiled with squares i ff a/b ∈ Q . Wednesday, May 11, 16

Thm(Dehn 1903): An a × b rectangle can be tiled with squares i ff a/b ∈ Q . 1 1 1 1 2 Wednesday, May 11, 16

alternate proof a + d = 1 a + f − d = 0 d e a − b − f = 0 d + f − e = 0 f b + c − f − e = 0 a b c b − c = 0       a 1 0 0 1 0 0 1 b 1 0 0 − 1 0 1 0             c 1 − 1 0 0 0 − 1 0       det K =?     =   d 0 0 0 1 − 1 1 0             e 0 1 1 0 − 1 − 1 0       f 0 1 − 1 0 0 0 0 K is a signed adjacency matrix of an underlying planar graph... Wednesday, May 11, 16

A t-graph in a polygon is a union of noncrossing line segments in which every endpoint lies on another segment, or on the boundary, or at a point where three or more segments meet, with one in each halfspace. a t-graph with four segments A t-graph is generic if no two endpoints are equal. Note: faces are convex. For generic t-graphs, 1 = χ (open disk) = #(faces) − #(segments) . Wednesday, May 11, 16

local pictures: generic not allowed nongeneric nongeneric generic not allowed (if three or more segments meet at a point, there must be one in each halfspace) generic nongeneric Wednesday, May 11, 16

Associated to a t-graph is a bipartite graph... Wednesday, May 11, 16

...which has dimer covers (when we remove all but one outer edge). Wednesday, May 11, 16

Wednesday, May 11, 16

(follows from [K-She ffi eld 2003]) Thm: The space of t-graphs with n segments, fixed boundary and fixed combinatorics is homeomorphic to R 2 n . Global coordinates are biratio coordinates { X i } . X ← − a − → → ← − d − → → − b − − X c a ← f − ← b d e c X = ac X = ace bd bd f Wednesday, May 11, 16

At a degenerate vertex, biratios are defined by continuity: c 1 X c 3 a 1 θ 1 θ 2 θ 3 a 2 Z Y b 3 b 2 X = c 1 sin θ 3 Y = a 2 sin θ 1 Z = b 3 sin θ 2 a 1 sin θ 2 b 2 sin θ 3 c 3 sin θ 1 Wednesday, May 11, 16

Proof idea: Let K be a Kasteleyn matrix with face weights X . Find diagonal matrices D W , D B such that   1 (1 , . . . , 1) D W KD B = 0 } . . D W KD B  = 0   .  except on boundary. 1 Use maximum principle to show embedding. ⇤ X ← − a − → → ← − d − → → − b − − X = ac c ← − ← bd Wednesday, May 11, 16

There are a number of special cases where one restricts the set of biratios. Wednesday, May 11, 16

Special case 1. Convex embeddings of graphs An embedding of a graph in R 2 is convex if its faces are convex Thm: The space of convex embeddings of G (with pinned boundary) is homeomorphic to R 2 V . Wednesday, May 11, 16

Proof: Take a nearby nondegenerate t-graph and set products of biratios around “vertices” to be 1. Show that any such assignment of biratios results in an embedding. ⇤ c c X a θ 1 θ 2 θ 3 a Z Y b b X = c sin θ 3 Y = a sin θ 1 Z = b sin θ 2 Note XY Z = 1 a sin θ 2 b sin θ 3 c sin θ 1 Wednesday, May 11, 16

Proof: Take a nearby nondegenerate t-graph and set products of biratios around “vertices” to be 1. Show that any such assignment of biratios results in an embedding. ⇤ c c X a θ 1 θ 2 θ 3 a Z Y b b X = c sin θ 3 Y = a sin θ 1 Z = b sin θ 2 Note XY Z = 1 a sin θ 2 b sin θ 3 c sin θ 1 note that X, Y, Z are ratios of barycentric coordinates! Wednesday, May 11, 16

A natural probability measure on convex embeddings is obtained by choosing transition probabilities iid in { 0 ≤ p, q, p + q ≤ 1 } . Wednesday, May 11, 16

Special case 2. Product of X s around both faces and vertices is 1. One can show that these conditions correspond to harmonic embeddings (spring networks / resistor networks) Wednesday, May 11, 16

Random convex embedding Random harmonic embedding Wednesday, May 11, 16

A random convex embedding does not have a scaling limit shape. Conjecture: Conjecture [Zeitouni]: A random convex embedding has a scaling limit shape. (would follow from CLT for RWRE) Wednesday, May 11, 16

Q. Is there a natural probability measure on Homeo( D 2 , D 2 )? Wednesday, May 11, 16

(Fix exact shapes up to scale) Special case 3. discrete analytic functions e.g. square tilings (all X s equal to 1) Wednesday, May 11, 16

Discrete analytic functions y 2 x 2 y 1 y 2 x 1 y 1 x 2 x 1 y 2 − y 1 = c ( x 2 − x 1 ) “discrete Cauchy-Riemann” f x = g y f y = − g x Wednesday, May 11, 16

More generally K is a discrete version of ∂ ¯ z e.g. regular hexagons and equilateral triangles (all X ’s equal to 1.) Wednesday, May 11, 16

(product of adjacent X s is 1) Rectangle tilings (square young tableau limit shape) Wednesday, May 11, 16

Fixed areas: Given a rectangle tiling, there is an “isotopic” rectangle tiling with prescribed areas. 6 7 1 ⇣ ⌘ 1 √ 19 + 73 36 5 4 4 x/y = 1 xy = 1 / 6 Wednesday, May 11, 16

Thm [K-Abrams] For every bipolar orientation of a planar graph, there is a unique Smith diagram with area-1 rectangles; that is, there is a unique choice of conductances so that the associated harmonic function has energies 1 and that orientation. Bipolar orientation: Acyclic with exactly one source and sink (on outer boundary). Wednesday, May 11, 16

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A random bipolar orientation of a random graph: Wednesday, May 11, 16

thank you for your attention! Wednesday, May 11, 16

Wednesday, May 11, 16

Wednesday, May 11, 16

Wednesday, May 11, 16

K : R W → R B signed (weighted) adjacency matrix Thm[Kasteleyn(1965)]: X det K = wt ( m ) . dimer covers m Q. What is the geometry underlying K ? Wednesday, May 11, 16