Counting colored maps: algebraicity results ArXiv: 0909.1695 Olivier Bernardi, MIT Joint work with Mireille Bousquet-Mélou IHP 2009 IHP 2009 Olivier Bernardi – p.1/25
Outline 1. Potts polynomial. 2. Functional equation for Potts model (easy part). 3. Solving equations (hard part). 4. Results and open questions. IHP 2009 Olivier Bernardi – p.2/25
Potts polynomial IHP 2009 Olivier Bernardi – p.3/25
Potts model A q -coloring of G = ( V, E ) is a function c : V �→ { 1 , 2 , . . . , q } . m ( c ) = 2 An edge is monochromatic if its endpoints have the same color. IHP 2009 ▽ Olivier Bernardi – p.4/25
Potts model A q -coloring of G = ( V, E ) is a function c : V �→ { 1 , 2 , . . . , q } . m ( c ) = 2 The Potts polynomial (partition function of the Potts model) is � u m ( c ) , P G ( q, u ) = c : V �→ [ q ] where m ( c ) is the number of monochromatic edges. IHP 2009 ▽ Olivier Bernardi – p.4/25
Potts model A q -coloring of G = ( V, E ) is a function c : V �→ { 1 , 2 , . . . , q } . m ( c ) = 2 The Potts polynomial (partition function of the Potts model) is � u m ( c ) , P G ( q, u ) = c : V �→ [ q ] where m ( c ) is the number of monochromatic edges. Remark: The chromatic polynomial P G ( q, 0) counts proper colorings. IHP 2009 Olivier Bernardi – p.4/25
Potts polynomial Fact: The Potts polynomial P G ( q, u ) = � c : V �→ [ q ] u m ( c ) , is a polynomial in q, u satisfying : P G ( q, u ) = P G \ e ( q, u ) + ( u − 1) P G/e ( q, u ) . G \ e Deletion G e G /e Contraction IHP 2009 Olivier Bernardi – p.5/25
Potts polynomial Fact: [Fortuin and Kastelein 72] The Potts polynomial and Tutte polynomial are equivalent. IHP 2009 ▽ Olivier Bernardi – p.6/25
Potts polynomial Fact: [Fortuin and Kastelein 72] The Potts polynomial and Tutte polynomial are equivalent. u m ( c ) = � � � (1 + δ ( c i , c j )( u − 1)) c : V �→ [ q ] c : V �→ [ q ] ( i,j ) ∈ E � � � = δ ( c i , c j )( u − 1) S ⊆ E c : V �→ [ q ] ( i,j ) ∈ S � � � = δ ( c i , c j )( u − 1) S ⊆ E c : V �→ [ q ] ( i,j ) ∈ S � q k ( S ) ( u − 1) | S | = S ⊆ E where k ( S ) is the number of connected components. IHP 2009 ▽ Olivier Bernardi – p.6/25
Potts polynomial Fact: [Fortuin and Kastelein 72] The Potts polynomial and Tutte polynomial are equivalent. Remarks: • The Potts model of a planar graph G and of its dual graph G ∗ are related (by P G ∗ ( q, u ) = ( u − 1) e ( G ) q v ( G ) − 1 P G ( q, 1 + q/ ( u − 1)) ). • the Potts polynomial can be specialized to count various structures: spanning trees, forests, connected subgraphs, acyclic orientations, score vectors, bipolar orientations, sandpile configurations... IHP 2009 Olivier Bernardi – p.6/25
Maps A planar map is an embedding of a connected planar graph in the sphere, considered up to continuous deformation. � = = (I indicate the rooting by pointing a corner) IHP 2009 Olivier Bernardi – p.7/25
Potts model on Maps The partition function of the (annealed) Potts model on maps is � P M ( q, u ) z | M | . G ( q, u, z ) = M map Phase transitions can be characterized by analyzing the singularities of G ( q, u, z ) . IHP 2009 ▽ Olivier Bernardi – p.8/25
Potts model on Maps The partition function of the (annealed) Potts model on maps is � P M ( q, u ) z | M | . G ( q, u, z ) = M map Remark: The series G ( q, u, z ) contains (as specializations) • the GF of maps G (1 , 1 , z ) , • the GF of properly q -colored maps G ( q, 0 , z ) , • the GF of tree-rooted maps (spanning trees), • the GF of Baxter numbers (bipolar orientations),... IHP 2009 ▽ Olivier Bernardi – p.8/25
Potts model on Maps The partition function of the (annealed) Potts model on maps is � P M ( q, u ) z | M | . G ( q, u, z ) = M map Question: For which values of q, u is G ( q, u, z ) algebraic ? (meaning P ( G ( q, u, z ) , z ) = 0 for a polynomial P � = 0 ) IHP 2009 ▽ Olivier Bernardi – p.8/25
Potts model on Maps The partition function of the (annealed) Potts model on maps is � P M ( q, u ) z | M | . G ( q, u, z ) = M map Question: For which values of q, u is G ( q, u, z ) algebraic ? Known: • GF of maps is algebraic [Tutte]. • GF of tree-rooted maps or Baxter numbers are not algebraic. • GF of properly colored triangulation T ( q, 0 , z ) is algebraic for q = 2 + 2 cos(2 π/m ) [Tutte / Richmond, Odlyzko 83]. • Results in [Bonnet, Eynard 99] suggests that T ( q, u, z ) is algebraic for q = 2 + 2 cos( kπ/m ) . IHP 2009 ▽ Olivier Bernardi – p.8/25
Potts model on Maps The partition function of the (annealed) Potts model on maps is � P M ( q, u ) z | M | . G ( q, u, z ) = M map Question: For which values of q, u is G ( q, u, z ) algebraic ? Thm [B., MBM]: The GF G ( q, u, z ) of the Potts model on planar maps is algebraic for q � = 0 , 4 of the form q = 2 + 2 cos( kπ/m ) . The same is true for the GF concerning triangulations. √ √ Examples : q = 1 , 2 , 3 , 2 + 2 , 2 + 3 ... IHP 2009 Olivier Bernardi – p.8/25
Functional equations for colored maps (a.k.a. loop equations ) IHP 2009 Olivier Bernardi – p.9/25
Generatingfunctionology Class A (+size function) → Generating function z | A | = � � a n z n . A ( z ) = A ∈A n ≥ 0 Recursive description of A → Equation for A ( z ) IHP 2009 ▽ Olivier Bernardi – p.10/25
Generatingfunctionology Class A (+size function) → Generating function z | A | = � � a n z n . A ( z ) = A ∈A n ≥ 0 Recursive description of A → Equation for A ( z ) Combinatorial description → generating function C ( z ) = A ( z ) + B ( z ) Disjoint union C = A ⊎ B → C ( z ) = A ( z ) × B ( z ) Cartesian product C = A × B → 1 C ( z ) = Sequence C = Seq ( A ) → 1 − A ( z ) . . . . . . IHP 2009 Olivier Bernardi – p.10/25
Example: plane trees � a n z n . Generating function of rooted plane trees: A ( z ) = n = ⊎ A ( z ) = 1 + z A ( z ) 2 . → IHP 2009 ▽ Olivier Bernardi – p.11/25
Example: plane trees � a n z n . Generating function of rooted plane trees: A ( z ) = n = ⊎ A ( z ) = 1 + z A ( z ) 2 . → The GF of plane trees is algebraic ! More generally, classes of trees defined by (finite) degree constraints are algebraic. IHP 2009 Olivier Bernardi – p.11/25
Recursive description for maps [Tutte 63] � z e ( M ) . G ( z ) = M ∈M = + + IHP 2009 ▽ Olivier Bernardi – p.12/25
Recursive description for maps [Tutte 63] � z e ( M ) . G ( z ) = M ∈M = + + G ( z ) = 1 + IHP 2009 ▽ Olivier Bernardi – p.12/25
Recursive description for maps [Tutte 63] � z e ( M ) . G ( z ) = M ∈M = + + G ( z ) = 1 + zG ( z ) 2 + IHP 2009 ▽ Olivier Bernardi – p.12/25
Recursive description for maps [Tutte 63] � z e ( M ) . G ( z ) = M ∈M = + + G ( z ) = 1 + zG ( z ) 2 + ? We are forced to take the degree of the root-face d f into account. IHP 2009 ▽ Olivier Bernardi – p.12/25
Recursive description for maps [Tutte 63] � x d f ( M ) z e ( M ) . G ( x, z ) = M ∈M = + + � yG ( y, z ) − G (1 , z ) � G ( y, z ) = 1 + y 2 zG ( y, z ) 2 + yz . y − 1 A small map M corresponds to from d f ( M ) + 1 big maps � � xz n + . . . + x k +1 z n = xz n x k +1 − 1 x k z n − 1 ❀ x − 1 IHP 2009 ▽ Olivier Bernardi – p.12/25
Recursive description for maps [Tutte 63] � x d f ( M ) z e ( M ) . G ( x, z ) = M ∈M = + + Remarks: • To describe maps by root-deletion we were forced to record the root-face degree. • To describe maps by root-contraction we would be forced to record the root-vertex degree. IHP 2009 Olivier Bernardi – p.12/25
Equation for Potts model on maps [Tutte 71] v ( M ) z e ( M ) P M ( q, u ) � x d f ( M ) y d G ( x, y ) ≡ . q M ∈M = + + + IHP 2009 ▽ Olivier Bernardi – p.13/25
Equation for Potts model on maps [Tutte 71] v ( M ) z e ( M ) P M ( q, u ) � x d f ( M ) y d G ( x, y ) ≡ . q M ∈M = + + + G ( x, y ) = 1 + ( q − 1+ u ) x 2 yzG ( x, y ) G ( x, 1) + uxy 2 zG ( x, y ) G (1 , y ) � � � � xG ( x,y ) − G (1 ,y ) + xyz − xyzG ( x, y ) G (1 , y ) x − 1 � � � � xG ( x,y ) − G ( x, 1) +( u − 1) xyz − xyzG ( x, y ) G ( x, 1) . y − 1 IHP 2009 Olivier Bernardi – p.13/25
Other equations Properly colored triangulations [Tutte 73]: T ( x, y ) = ( q − 1) y + xyz T ( x, y ) T ( x, 1) + yz T ( x, y ) − T (0 , y ) − xy 2 z T ( x, y ) − T ( x, 1) . x y − 1 Potts model on cubic maps [Eynard, Bonnet 99]: T ( x, y ) − T 0 ( y ) − T ( x, y ) − T 0 ( x ) + ( u − 1) z ( x T 0 ( x ) − y T 0 ( y )) T ( x, y ) x y „ T ( x, y ) − T 0 ( x ) − y T 1 ( x ) − T ( x, y ) − T 0 ( y ) − x T 1 ( xy ) « = ( u − 1) z . y 2 x 2 Alternatively, Potts model on triangulations [B., MBM]: T ( x, y ) = 1 + x 2 z ( q + u − 1) T ( x, y ) T ( x, 0) + uxz ( T 2 ( y ) + 2 y T 1 ( y )) T ( x, y ) T ( x, y ) − 1 − x T 1 ( y ) − x 2 T ( x, y ) T 2 ( y ) ` ´ + yz ( T ( x, y ) − 1 − x T 1 ( y ) T ( x, y )) + z x + x 2 z 2 ( u − 1) yu T ( x, y ) T ( x, 0) + xz ( u − 1) ( T ( x, y ) − T ( x, 0)) . 1 − yuz (1 − yuz ) y There exists equations for Potts model on p -angulations for any p [B., MBM]. IHP 2009 Olivier Bernardi – p.14/25
Solving functional equations IHP 2009 Olivier Bernardi – p.15/25
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