Miquel dynamics for circle patterns
Sanjay Ramassamy
École normale supérieure
Includes joint works with:Alexey Glutsyuk (ENS Lyon / HSE Moscow)
Richard Kenyon (Brown University)
Wai Yeung Lam (Brown University)
Marianna Russkikh (Université de Genève)
13th conference on
Symmetries and Integrability of Difference Equations
Fukuoka, November 15 2018• Circle patterns form a well-studied class of objects indiscrete differential geometry (discretization of conformal maps).

• Miquel dynamics: discrete integrable system on thespace of circle patterns.

• Connection between circle patterns and the dimermodel.

Miquel’s theorem
A0A
B
B0DD0CC0Miquel’s theorem
A0AB
B0DD0CC0Theorem (Miquel, 1838). In this setting,A, B, C, D concyclic ⇔ A0 , B 0 , C 0 , D0 concyclic.

Square grid circle patterns
Definition. A map S : Z2 → R2 is called a square grid circle
pattern if any four vertices around a face of Z2 get mapped
to four concyclic points.

SSquare grid circle patterns
Definition. A map S : Z2 → R2 is called a square grid circle
pattern if any four vertices around a face of Z2 get mapped
to four concyclic points.

S

Miquel dynamics• Checkerboard coloring of the faces of Z2 : black and whitecircles.

• Define two maps from the set of square grid circle patterns
to itself, black mutation µB and white mutation µW .

• Black mutation µB : each vertex gets moved to the otherintersection point of the two white circles it belongs to.
All the vertices move simultaneously.

• Black mutation µB : each vertex gets moved to the otherintersection point of the two white circles it belongs to.
All the vertices move simultaneously.

• Black mutation µB : each vertex gets moved to the otherintersection point of the two white circles it belongs to.
All the vertices move simultaneously.

• Black mutation µB : each vertex gets moved to the otherintersection point of the two white circles it belongs to.
All the vertices move simultaneously.

• Black mutation µB : each vertex gets moved to the otherintersection point of the two white circles it belongs to.
All the vertices move simultaneously.

• Why does µB produce a square grid circle pattern ?

• Black mutation µB : each vertex gets moved to the otherintersection point of the two white circles it belongs to.
All the vertices move simultaneously.

• Why does µB produce a square grid circle pattern ?

• Black mutation µB : each vertex gets moved to the otherintersection point of the two white circles it belongs to.
All the vertices move simultaneously.

Miquel’s theorem !

• Why does µB produce a square grid circle pattern ?

• The maps µB and µW are involutions.

• Miquel dynamics: discrete-time dynamics obtained byalternating between µB and µW .

• Invented by Richard Kenyon.

• Resembles (but a priori unrelated to) the dynamics on
circle configurations in three dimensions considered by
Bazhanov-Mangazeev-Sergeev.

Biperiodic square grid circle patterns• A circle pattern S is spatially biperiodic if there exist
m, n integers and ~u, ~v ∈ R2 such that for all (x, y) ∈ R2 ,S(x + m, y) = S(x, y) + ~uS(x, y + n) = S(x, y) + ~v

m=4n=2nm

• A biperiodic circle pattern is mapped by Miquel dynamics to another biperiodic circle pattern with the sameperiods (m, n) and the same monodromies (~u, ~v ).

• This reduces the problem to a finite-dimensional one.

• A biperiodic circle pattern in the plane projects down
to a circle pattern on a flat torus.

[Mathematica]

Towards integrability
m=4n=2
Theorem (R., 2018). For m × n biperiodic SGCPs, the sumof the turning angles along a zigzag loop is invariant underMiquel dynamics.
Theorem (Glutsyuk-R., 2018). For 2 × 2 biperiodic SGCPs,the relative motion under Miquel dynamics of a vertex withrespect to another corresponds to translation on an elliptic
curve.

Towards integrability
m=4n=2
Theorem (R., 2018). For m × n biperiodic SGCPs, the sumof the turning angles along a zigzag loop is invariant underMiquel dynamics.
Theorem (Glutsyuk-R., 2018). For 2 × 2 biperiodic SGCPs,
the relative motion under Miquel dynamics of a vertex withrespect to another corresponds to translation on an elliptic
curve.

Towards integrability
m=4n=2
Theorem (R., 2018). For m × n biperiodic SGCPs, the sumof the turning angles along a zigzag loop is invariant underMiquel dynamics.
Theorem (Glutsyuk-R., 2018). For 2 × 2 biperiodic SGCPs,
the relative motion under Miquel dynamics of a vertex withrespect to another corresponds to translation on an ellipticcurve.

Key observation: study the evolutionof circle centers rather than vertices(intersection points).
[Geogebra]

• Two-parameter family of circle patterns with given circle centers (pick one vertex freely).

Circle pattern 1samecenters

black/whitemutation
Circle pattern 10Circle pattern 2
black/whitemutationsame
centers

Circle pattern 20• Miquel dynamics induces a dynamics on circle centers.

Miquel dynamics on centersMiquel dynamics on centersMiquel dynamics on centersnw

c0e
cn, w, s, e, c, c0 ∈ CsTheorem (Affolter 2018, Kenyon-Lam-R.-Russkikh 2018).
(c − w)(s − c0 )(e − n)
= −10
(w − s)(c − e)(n − c)(c0,0,−1 −c−1,0,0 )(c0,−1,0 −c0,0,1 )(c1,0,0 −c0,1,0 )= −1(c−1,0,0 −c0,−1,0 )(c0,0,1 −c1,0,0 )(c0,1,0 −c0,0,−1 )

c0,1,0c0,0,1c−1,0,0
c1,0,0c0,0,−1c0,−1,0• Cauchy initial data: positions of the white centers attime −1 and of the black centers at time 0.
• Compute iteratively the positions of all centers using
the above equation on each elementary octahedron.

The discrete Schwarzian KP equation(c0,0,−1 −c−1,0,0 )(c0,−1,0 −c0,0,1 )(c1,0,0 −c0,1,0 )= −1(c−1,0,0 −c0,−1,0 )(c0,0,1 −c1,0,0 )(c0,1,0 −c0,0,−1 )

c0,1,0c0,0,1c−1,0,0
c1,0,0c0,0,−1c0,−1,0• Cauchy initial data: positions of the white centers attime −1 and of the black centers at time 0.
• Compute iteratively the positions of all centers using
the above equation on each elementary octahedron.

The discrete Schwarzian KP equation(c0,0,−1 −c−1,0,0 )(c0,−1,0 −c0,0,1 )(c1,0,0 −c0,1,0 )= −1(c−1,0,0 −c0,−1,0 )(c0,0,1 −c1,0,0 )(c0,1,0 −c0,0,−1 )• Governs discrete Laplace-Darboux transformations(Doliwa) and the pentagram map (Schief).
• Geometric interpretation given by Clifford’s circle theorem (Konopelchenko-Schief).
• Equation χ2 in the Adler-Bobenko-Suris classificationof integrable equations of octahedron type.

X variables for circle centers...
• Given centers of a circle pattern, associate an X variable to each face.

nw

cse

(c−n)(c−s)
X = − (c−w)(c−e)

... evolve like a Y -system
XnXw

X
Xs

blackmutation

X 7−→ X −1

whitemutation(1+Xn )(1+Xs )−7→XX(1+X −1 )(1+X −1 )Xewe• Transformation rule for coefficient variables in a cluster
algebra.

The dimer model• Statistical mechanical model for random perfect matchings on graphs.
• Setting: planar graph with all faces of even degree.
Each edge carries a positive real weight.
acbedfgDimer X variables• Given a graph with edge weights, associate an X variable to each face by taking the alternating product ofedge weights around the face.
abdac
X = bdc• If two collections of edge weights on a graph induce thesame X variables on the faces, then the two collectionsof edge weights define the same probabilistic model.

Dimer local moves• Two local moves preserving the probabilistic model.
• Contraction of degree 2 vertices:11• Urban renewal:
ab1d/∆d∆ = ac + bdcc/∆

1

1b/∆

a/∆

1

Goncharov-Kenyon dimer dynamics• Start with Z2 with some edge weights. At even (resp.
odd) times, perform an urban renewal on each white(resp. black) face followed by the contraction of all thedegree 2 vertices. Get Z2 with different edge weights.

Goncharov-Kenyon dimer dynamics• Start with Z2 with some edge weights. At even (resp.
odd) times, perform an urban renewal on each white(resp. black) face followed by the contraction of all thedegree 2 vertices. Get Z2 with different edge weights.

Goncharov-Kenyon dimer dynamics• Start with Z2 with some edge weights. At even (resp.
odd) times, perform an urban renewal on each white(resp. black) face followed by the contraction of all thedegree 2 vertices. Get Z2 with different edge weights.

Goncharov-Kenyon dimer dynamics• Start with Z2 with some edge weights. At even (resp.
odd) times, perform an urban renewal on each white(resp. black) face followed by the contraction of all thedegree 2 vertices. Get Z2 with different edge weights.

Goncharov-Kenyon dimer dynamics• Start with Z2 with some edge weights. At even (resp.
odd) times, perform an urban renewal on each white(resp. black) face followed by the contraction of all thedegree 2 vertices. Get Z2 with different edge weights.

Goncharov-Kenyon dimer dynamics• Start with Z2 with some edge weights. At even (resp.
odd) times, perform an urban renewal on each white(resp. black) face followed by the contraction of all thedegree 2 vertices. Get Z2 with different edge weights.
• The dimer X variables evolve like a Y -system.
XnXwX
Xs

black
urban
renewals

X 7−→ X −1whiteurban
renewals(1+Xn )(1+Xs )−7→X
X(1+X −1 )(1+X −1 )Xewe

From circle centers to dimers• Starting from a circle pattern realization of a graph Gwith positive X variables, define the dimer weight oneach edge of G as the distance between the two centersseparated by the edge.

• Extends the map defined in the isoradial case (all thecircles have the same radius) by Kenyon in 2002.

circle centers
black/whitemutation
circle centers

dimer model
black/whiteurban renewals
dimer model

Theorem (Affolter 2018, Kenyon-Lam-R.-Russkikh 2018).
The diagram commutes in the sense that the two dimer models produced by aaaaa and aaaaa have the same X variables.

• Isoradial case: the two dimers models produced byaaaaa and aaaaa have the same edge weights [R. 2018].

From dimers to circle centers• Impossible to recover the circle centers from the dimer
model in the infinite planar case.
• To a dimer model on the torus (infinite biperiodic case),
associate its spectral curve (algebraic curve).
Theorem (Kenyon-Lam-R.-Russkikh 2018). For graphs on
the torus, the correspondence(dimer model, point on its spectral curve) ↔ {circle centers}
is bijective if we quotient out by the appropriate symmetries.

Miquel’s integrals of motion• The integrals of motion of the dimer dynamics on thetorus are given by the coefficients of the polynomial
defining the spectral curve (Goncharov-Kenyon).
• Integrals of motion of Miquel dynamics on biperiodic
circle centers: dimer integrals of motion together withthe two sums of turning angles along zigzag paths.

N. Affolter, Miquel dynamics, Clifford lattices and the dimermodel, arXiv:1808.04227 (2018).
A. Glutsyuk and S. Ramassamy, A first integrability resultfor Miquel dynamics, J. Geom. Phys., 130 (2018), 121-129.
R. Kenyon, W. Y. Lam, S. Ramassamy and M. Russkikh,Dimers and circle patterns, arXiv:1810.05616 (2018).
S. Ramassamy, Miquel dynamics for circle patterns, Int.
Math. Res. Not. (2018), published electronically.

THANK YOU !