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Ising model and total positivity Pavel Galashin MIT galashin@mit.edu University of Toronto Colloquium, January 7, 2019 Joint work with Pavlo Pylyavskyy arXiv:1807.03282 Pavel Galashin (MIT) Ising model and total positivity U of Toronto,

  1. Ising model: history Suggested by by W. Lenz to his student E. Ising in 1920 Ising (1925): not a good model for ferromagnetism Q: how does | � F | depend on T ◦ ? | � F | T ◦ � F | � F | T ◦ Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 6 / 29

  2. Ising model: history Suggested by by W. Lenz to his student E. Ising in 1920 Ising (1925): not a good model for ferromagnetism Q: how does | � F | depend on T ◦ ? | � F | T ◦ � F | � F | T ◦ Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 6 / 29

  3. Ising model: history Suggested by by W. Lenz to his student E. Ising in 1920 Ising (1925): not a good model for ferromagnetism Q: how does | � F | depend on T ◦ ? | � F | T ◦ � F | � F | T ◦ Curie point (P. Curie, 1895) Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 6 / 29

  4. Ising model: history Suggested by by W. Lenz to his student E. Ising in 1920 Ising (1925): not a good model for ferromagnetism Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 7 / 29

  5. Ising model: history Suggested by by W. Lenz to his student E. Ising in 1920 ⇒ not a good model for Ising (1925): no phase transition in 1D = ferromagnetism Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 7 / 29

  6. Ising model: history Suggested by by W. Lenz to his student E. Ising in 1920 ⇒ not a good model for Ising (1925): no phase transition in 1D = ferromagnetism Historically, we let G := Z d ∩ Ω for some Ω ⊂ R d and set all J e := 1 T for some temperature T ∈ R > 0 . Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 7 / 29

  7. Ising model: history Suggested by by W. Lenz to his student E. Ising in 1920 ⇒ not a good model for Ising (1925): no phase transition in 1D = ferromagnetism Historically, we let G := Z d ∩ Ω for some Ω ⊂ R d and set all J e := 1 T for some temperature T ∈ R > 0 . Peierls (1937): phase transition in Z d for d ≥ 2 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 7 / 29

  8. Ising model: history Suggested by by W. Lenz to his student E. Ising in 1920 ⇒ not a good model for Ising (1925): no phase transition in 1D = ferromagnetism Historically, we let G := Z d ∩ Ω for some Ω ⊂ R d and set all J e := 1 T for some temperature T ∈ R > 0 . Peierls (1937): phase transition in Z d for d ≥ 2 � √ T c = 1 1 � for Z 2 Kramers–Wannier (1941): critical temperature 2 log 2 + 1 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 7 / 29

  9. Ising model: phase transition Let G ⊂ Z d be a (2 N + 1) × (2 N + 1) square and J e = 1 T for some fixed T ∈ R > 0 . Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 8 / 29

  10. Ising model: phase transition Let G ⊂ Z d be a (2 N + 1) × (2 N + 1) square and J e = 1 T for some fixed T ∈ R > 0 . Suppose that σ u = +1 for all u ∈ ∂ G . Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 8 / 29

  11. Ising model: phase transition Let G ⊂ Z d be a (2 N + 1) × (2 N + 1) square and J e = 1 T for some fixed T ∈ R > 0 . Suppose that σ u = +1 for all u ∈ ∂ G . v Let v be the vertex in the middle of the square. Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 8 / 29

  12. Ising model: phase transition Let G ⊂ Z d be a (2 N + 1) × (2 N + 1) square and J e = 1 T for some fixed T ∈ R > 0 . Suppose that σ u = +1 for all u ∈ ∂ G . v Let v be the vertex in the middle of the square. Define the spontaneous magnetization N →∞ (Prob( σ v = +1) − Prob( σ v = − 1)) M ( T ) := lim Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 8 / 29

  13. Ising model: phase transition Let G ⊂ Z d be a (2 N + 1) × (2 N + 1) square and J e = 1 T for some fixed T ∈ R > 0 . Suppose that σ u = +1 for all u ∈ ∂ G . v Let v be the vertex in the middle of the square. Define the spontaneous magnetization N →∞ (Prob( σ v = +1) − Prob( σ v = − 1)) M ( T ) := lim Theorem (Onsager (1944),Onsager–Kaufman (1949), Yang (1952)) M ( T ) T c T Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 8 / 29

  14. Ising model: phase transition Let G ⊂ Z d be a (2 N + 1) × (2 N + 1) square and J e = 1 T for some fixed T ∈ R > 0 . Suppose that σ u = +1 for all u ∈ ∂ G . v Let v be the vertex in the middle of the square. Define the spontaneous magnetization N →∞ (Prob( σ v = +1) − Prob( σ v = − 1)) M ( T ) := lim Theorem (Onsager (1944),Onsager–Kaufman (1949), Yang (1952)) 1 M ( T ) ≍ ( T c − T ) 8 T c T Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 8 / 29

  15. Ising model: history Suggested by by W. Lenz to his student E. Ising in 1920 Ising (1925): no phase transition in 1D = ⇒ not a good model for ferromagnetism Historically, we let G := Z d ∩ Ω for some Ω ⊂ R d and set all J e := 1 T for some temperature T ∈ R > 0 . Peierls (1937): phase transition in Z d for d ≥ 2 � √ T c = 1 1 for Z 2 � Kramers–Wannier (1941): critical temperature 2 log 2 + 1 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 9 / 29

  16. Ising model: history Suggested by by W. Lenz to his student E. Ising in 1920 Ising (1925): no phase transition in 1D = ⇒ not a good model for ferromagnetism Historically, we let G := Z d ∩ Ω for some Ω ⊂ R d and set all J e := 1 T for some temperature T ∈ R > 0 . Peierls (1937): phase transition in Z d for d ≥ 2 � √ T c = 1 1 for Z 2 � Kramers–Wannier (1941): critical temperature 2 log 2 + 1 Onsager, Kaufman, Yang (1944–1952): exact expressions for the free energy and spontaneous magnetization Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 9 / 29

  17. Ising model: history Suggested by by W. Lenz to his student E. Ising in 1920 Ising (1925): no phase transition in 1D = ⇒ not a good model for ferromagnetism Historically, we let G := Z d ∩ Ω for some Ω ⊂ R d and set all J e := 1 T for some temperature T ∈ R > 0 . Peierls (1937): phase transition in Z d for d ≥ 2 � √ T c = 1 1 for Z 2 � Kramers–Wannier (1941): critical temperature 2 log 2 + 1 Onsager, Kaufman, Yang (1944–1952): exact expressions for the free energy and spontaneous magnetization Belavin–Polyakov–Zamolodchikov (1984): conjectured conformal invariance of the scaling limit at T = T c for Z 2 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 9 / 29

  18. Ising model: history Suggested by by W. Lenz to his student E. Ising in 1920 Ising (1925): no phase transition in 1D = ⇒ not a good model for ferromagnetism Historically, we let G := Z d ∩ Ω for some Ω ⊂ R d and set all J e := 1 T for some temperature T ∈ R > 0 . Peierls (1937): phase transition in Z d for d ≥ 2 � √ T c = 1 1 for Z 2 � Kramers–Wannier (1941): critical temperature 2 log 2 + 1 Onsager, Kaufman, Yang (1944–1952): exact expressions for the free energy and spontaneous magnetization Belavin–Polyakov–Zamolodchikov (1984): conjectured conformal invariance of the scaling limit at T = T c for Z 2 Smirnov, Chelkak, Hongler, Izyurov, ... (2010–2015): proved conformal invariance and universality of the scaling limit at T = T c for Z 2 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 9 / 29

  19. Ising model: boundary correlations Recall: G is embedded in a disk. Let b 1 , . . . , b n be the boundary vertices. b 3 b 2 b 4 b 1 b 5 b 6 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 10 / 29

  20. Ising model: boundary correlations Recall: G is embedded in a disk. Let b 1 , . . . , b n be the boundary vertices. Correlation: � σ u σ v � := Prob( σ u = σ v ) − Prob( σ u � = σ v ). b 3 b 2 b 4 b 1 b 5 b 6 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 10 / 29

  21. Ising model: boundary correlations Recall: G is embedded in a disk. Let b 1 , . . . , b n be the boundary vertices. Correlation: � σ u σ v � := Prob( σ u = σ v ) − Prob( σ u � = σ v ). Definition Boundary correlation matrix: M ( G , J ) = ( m ij ) n i , j =1 , where m ij := � σ b i σ b j � . b 3 b 2 b 4 b 1 b 5 b 6 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 10 / 29

  22. Ising model: boundary correlations Recall: G is embedded in a disk. Let b 1 , . . . , b n be the boundary vertices. Correlation: � σ u σ v � := Prob( σ u = σ v ) − Prob( σ u � = σ v ). Definition Boundary correlation matrix: M ( G , J ) = ( m ij ) n i , j =1 , where m ij := � σ b i σ b j � . b 3 b 2 M ( G , J ) is a symmetric matrix b 4 b 1 b 5 b 6 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 10 / 29

  23. Ising model: boundary correlations Recall: G is embedded in a disk. Let b 1 , . . . , b n be the boundary vertices. Correlation: � σ u σ v � := Prob( σ u = σ v ) − Prob( σ u � = σ v ). Definition Boundary correlation matrix: M ( G , J ) = ( m ij ) n i , j =1 , where m ij := � σ b i σ b j � . b 3 b 2 M ( G , J ) is a symmetric matrix with 1’s on the diagonal b 4 b 1 b 5 b 6 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 10 / 29

  24. Ising model: boundary correlations Recall: G is embedded in a disk. Let b 1 , . . . , b n be the boundary vertices. Correlation: � σ u σ v � := Prob( σ u = σ v ) − Prob( σ u � = σ v ). Definition Boundary correlation matrix: M ( G , J ) = ( m ij ) n i , j =1 , where m ij := � σ b i σ b j � . b 3 b 2 M ( G , J ) is a symmetric matrix with 1’s on the diagonal and nonnegative entries b 4 b 1 b 5 b 6 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 10 / 29

  25. Ising model: boundary correlations Recall: G is embedded in a disk. Let b 1 , . . . , b n be the boundary vertices. Correlation: � σ u σ v � := Prob( σ u = σ v ) − Prob( σ u � = σ v ). Definition Boundary correlation matrix: M ( G , J ) = ( m ij ) n i , j =1 , where m ij := � σ b i σ b j � . b 3 b 2 M ( G , J ) is a symmetric matrix with 1’s on the diagonal and nonnegative entries b 4 b 1 Lives inside R ( n 2 ) b 5 b 6 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 10 / 29

  26. Ising model: boundary correlations Recall: G is embedded in a disk. Let b 1 , . . . , b n be the boundary vertices. Correlation: � σ u σ v � := Prob( σ u = σ v ) − Prob( σ u � = σ v ). Definition Boundary correlation matrix: M ( G , J ) = ( m ij ) n i , j =1 , where m ij := � σ b i σ b j � . b 3 b 2 M ( G , J ) is a symmetric matrix with 1’s on the diagonal and nonnegative entries b 4 b 1 Lives inside R ( n 2 ) b 5 b 6 X n := { M ( G , J ) | ( G , J ) is a planar Ising network with n boundary vertices } Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 10 / 29

  27. Ising model: boundary correlations Recall: G is embedded in a disk. Let b 1 , . . . , b n be the boundary vertices. Correlation: � σ u σ v � := Prob( σ u = σ v ) − Prob( σ u � = σ v ). Definition Boundary correlation matrix: M ( G , J ) = ( m ij ) n i , j =1 , where m ij := � σ b i σ b j � . b 3 b 2 M ( G , J ) is a symmetric matrix with 1’s on the diagonal and nonnegative entries b 4 b 1 Lives inside R ( n 2 ) b 5 b 6 X n := { M ( G , J ) | ( G , J ) is a planar Ising network with n boundary vertices } X n := closure of X n inside the space of n × n matrices. Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 10 / 29

  28. Boundary correlations: an example for n = 2 J e b 2 b 1 b 2 b 1 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 11 / 29

  29. Boundary correlations: an example for n = 2 J e b 2 b 1 b 2 b 1 � 1 � m 12 M ( G , J ) = m 12 1 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 11 / 29

  30. Boundary correlations: an example for n = 2 J e b 2 b 1 b 2 b 1 � 1 � m 12 = � σ 1 σ 2 � = 2 exp( J e ) − 2 exp( − J e ) m 12 M ( G , J ) = , m 12 1 2 exp( J e ) + 2 exp( − J e ) Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 11 / 29

  31. Boundary correlations: an example for n = 2 J e b 2 b 1 b 2 b 1 � 1 � m 12 = � σ 1 σ 2 � = 2 exp( J e ) − 2 exp( − J e ) m 12 M ( G , J ) = , m 12 1 2 exp( J e ) + 2 exp( − J e ) J e = 0 J e ∈ (0 , ∞ ) J e = ∞ m 12 = 0 m 12 ∈ (0 , 1) m 12 = 1 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 11 / 29

  32. Boundary correlations: an example for n = 2 J e b 2 b 1 b 2 b 1 � 1 � m 12 = � σ 1 σ 2 � = 2 exp( J e ) − 2 exp( − J e ) m 12 M ( G , J ) = , m 12 1 2 exp( J e ) + 2 exp( − J e ) J e = 0 J e ∈ (0 , ∞ ) J e = ∞ m 12 = 0 m 12 ∈ (0 , 1) m 12 = 1 We have X 2 ∼ = [0 , 1) and X 2 ∼ = [0 , 1]. Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 11 / 29

  33. Boundary correlations: an example for n = 2 J e b 2 b 1 b 2 b 1 � 1 � m 12 = � σ 1 σ 2 � = 2 exp( J e ) − 2 exp( − J e ) m 12 M ( G , J ) = , m 12 1 2 exp( J e ) + 2 exp( − J e ) J e = 0 J e ∈ (0 , ∞ ) J e = ∞ m 12 = 0 m 12 ∈ (0 , 1) m 12 = 1 We have X 2 ∼ = [0 , 1) and X 2 ∼ = [0 , 1]. X n is neither open nor closed inside R ( n 2 ). Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 11 / 29

  34. Boundary correlations: an example for n = 2 J e ∞ b 2 b 1 b 2 b 1 b 2 b 1 � 1 � m 12 = � σ 1 σ 2 � = 2 exp( J e ) − 2 exp( − J e ) m 12 M ( G , J ) = , m 12 1 2 exp( J e ) + 2 exp( − J e ) J e = 0 J e ∈ (0 , ∞ ) J e = ∞ m 12 = 0 m 12 ∈ (0 , 1) m 12 = 1 We have X 2 ∼ = [0 , 1) and X 2 ∼ = [0 , 1]. X n is neither open nor closed inside R ( n 2 ). X n is obtained from X n by allowing J e = ∞ (i.e., contracting edges). Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 11 / 29

  35. Part 2: Total positivity

  36. The totally nonnegative (TNN) Grassmannian Gr( k , n ) := { W ⊂ R n | dim( W ) = k } . Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 13 / 29

  37. The totally nonnegative (TNN) Grassmannian Gr( k , n ) := { W ⊂ R n | dim( W ) = k } . Gr( k , n ) := { k × n matrices of rank k } /(row operations) . Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 13 / 29

  38. The totally nonnegative (TNN) Grassmannian Gr( k , n ) := { W ⊂ R n | dim( W ) = k } . Gr( k , n ) := { k × n matrices of rank k } /(row operations) . Example: � 1 − 1 � 1 0 RowSpan ∈ Gr(2 , 4) 0 2 1 1 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 13 / 29

  39. The totally nonnegative (TNN) Grassmannian Gr( k , n ) := { W ⊂ R n | dim( W ) = k } . Gr( k , n ) := { k × n matrices of rank k } /(row operations) . Example: � 1 − 1 � 1 0 RowSpan ∈ Gr(2 , 4) 0 2 1 1 ucker coordinates: for I ⊂ [ n ] := { 1 , 2 , . . . , n } of size k , Pl¨ Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 13 / 29

  40. The totally nonnegative (TNN) Grassmannian Gr( k , n ) := { W ⊂ R n | dim( W ) = k } . Gr( k , n ) := { k × n matrices of rank k } /(row operations) . Example: � 1 − 1 � 1 0 RowSpan ∈ Gr(2 , 4) 0 2 1 1 ucker coordinates: for I ⊂ [ n ] := { 1 , 2 , . . . , n } of size k , Pl¨ ∆ I := k × k minor with column set I . Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 13 / 29

  41. The totally nonnegative (TNN) Grassmannian Gr( k , n ) := { W ⊂ R n | dim( W ) = k } . Gr( k , n ) := { k × n matrices of rank k } /(row operations) . Example: � 1 − 1 � 1 0 ∆ 13 = 1 ∆ 12 = 2 ∆ 14 = 1 RowSpan ∈ Gr(2 , 4) 0 2 1 1 ∆ 24 = 3 ∆ 34 = 1 ∆ 23 = 1 . ucker coordinates: for I ⊂ [ n ] := { 1 , 2 , . . . , n } of size k , Pl¨ ∆ I := k × k minor with column set I . Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 13 / 29

  42. The totally nonnegative (TNN) Grassmannian Gr( k , n ) := { W ⊂ R n | dim( W ) = k } . Gr( k , n ) := { k × n matrices of rank k } /(row operations) . Example: � 1 − 1 � 1 0 ∆ 13 = 1 ∆ 12 = 2 ∆ 14 = 1 RowSpan ∈ Gr(2 , 4) 0 2 1 1 ∆ 24 = 3 ∆ 34 = 1 ∆ 23 = 1 . ucker coordinates: for I ⊂ [ n ] := { 1 , 2 , . . . , n } of size k , Pl¨ ∆ I := k × k minor with column set I . Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 13 / 29

  43. The totally nonnegative (TNN) Grassmannian Gr( k , n ) := { W ⊂ R n | dim( W ) = k } . Gr( k , n ) := { k × n matrices of rank k } /(row operations) . Example: � 1 − 1 � 1 0 ∆ 13 = 1 ∆ 12 = 2 ∆ 14 = 1 RowSpan ∈ Gr(2 , 4) 0 2 1 1 ∆ 24 = 3 ∆ 34 = 1 ∆ 23 = 1 . ucker coordinates: for I ⊂ [ n ] := { 1 , 2 , . . . , n } of size k , Pl¨ ∆ I := k × k minor with column set I . Definition (Postnikov (2006)) The totally nonnegative Grassmannian is Gr ≥ 0 ( k , n ) := { W ∈ Gr( k , n ) | ∆ I ( W ) ≥ 0 for all I } . Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 13 / 29

  44. The totally nonnegative (TNN) Grassmannian Gr( k , n ) := { W ⊂ R n | dim( W ) = k } . Gr( k , n ) := { k × n matrices of rank k } /(row operations) . Example: � 1 − 1 � 1 0 ∆ 13 = 1 ∆ 12 = 2 ∆ 14 = 1 RowSpan ∈ Gr ≥ 0 (2 , 4) 0 2 1 1 ∆ 24 = 3 ∆ 34 = 1 ∆ 23 = 1 . ucker coordinates: for I ⊂ [ n ] := { 1 , 2 , . . . , n } of size k , Pl¨ ∆ I := k × k minor with column set I . Definition (Postnikov (2006)) The totally nonnegative Grassmannian is Gr ≥ 0 ( k , n ) := { W ∈ Gr( k , n ) | ∆ I ( W ) ≥ 0 for all I } . Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 13 / 29

  45. Example: Gr ≥ 0 (2 , 4) � 1 � 1 0 − 1 RowSpan ∈ Gr ≥ 0 (2 , 4) 0 2 1 1 ∆ 13 = 1 , ∆ 24 = 3 , ∆ 12 = 2 , ∆ 34 = 1 , ∆ 14 = 1 , ∆ 23 = 1 . Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 14 / 29

  46. Example: Gr ≥ 0 (2 , 4) � 1 � 1 0 − 1 RowSpan ∈ Gr ≥ 0 (2 , 4) 0 2 1 1 u 1 u 2 u 3 u 4 ∆ 13 = 1 , ∆ 24 = 3 , ∆ 12 = 2 , ∆ 34 = 1 , ∆ 14 = 1 , ∆ 23 = 1 . Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 14 / 29

  47. Example: Gr ≥ 0 (2 , 4) u 2 u 4 u 3 � 1 � 1 0 − 1 RowSpan ∈ Gr ≥ 0 (2 , 4) 0 2 1 1 u 1 u 2 u 3 u 4 u 1 ∆ 13 = 1 , ∆ 24 = 3 , ∆ 12 = 2 , ∆ 34 = 1 , ∆ 14 = 1 , ∆ 23 = 1 . Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 14 / 29

  48. Example: Gr ≥ 0 (2 , 4) u 2 u 4 u 3 � 1 � 1 0 − 1 RowSpan ∈ Gr ≥ 0 (2 , 4) 0 2 1 1 u 1 u 2 u 3 u 4 u 1 ∆ 13 = 1 , ∆ 24 = 3 , ∆ 12 = 2 , ∆ 34 = 1 , ∆ 14 = 1 , ∆ 23 = 1 . In Gr(2 , 4), we have a Pl¨ ucker relation: ∆ 13 ∆ 24 = ∆ 12 ∆ 34 + ∆ 14 ∆ 23 . Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 14 / 29

  49. Example: Gr ≥ 0 (2 , 4) u 2 u 4 u 3 � 1 � 1 0 − 1 RowSpan ∈ Gr ≥ 0 (2 , 4) 0 2 1 1 u 1 u 2 u 3 u 4 u 1 ∆ 13 = 1 , ∆ 24 = 3 , ∆ 12 = 2 , ∆ 34 = 1 , ∆ 14 = 1 , ∆ 23 = 1 . In Gr(2 , 4), we have a Pl¨ ucker relation: ∆ 13 ∆ 24 = ∆ 12 ∆ 34 + ∆ 14 ∆ 23 . Top cell: ∆ 13 , ∆ 24 , ∆ 12 , ∆ 34 , ∆ 14 , ∆ 23 > 0 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 14 / 29

  50. Example: Gr ≥ 0 (2 , 4) u 2 u 1 u 4 u 3 � 1 � 1 0 − 1 RowSpan ∈ Gr ≥ 0 (2 , 4) 0 2 1 1 u 1 u 2 u 3 u 4 ∆ 13 = 1 , ∆ 24 = 3 , ∆ 12 = 2 , ∆ 34 = 1 , ∆ 14 = 1 , ∆ 23 = 1 . In Gr(2 , 4), we have a Pl¨ ucker relation: ∆ 13 ∆ 24 = ∆ 12 ∆ 34 + ∆ 14 ∆ 23 . Top cell: ∆ 13 , ∆ 24 , ∆ 12 , ∆ 34 , ∆ 14 , ∆ 23 > 0 Codimension 1 cells: ∆ 12 = 0 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 14 / 29

  51. Example: Gr ≥ 0 (2 , 4) u 2 u 4 u 3 � 1 � 1 0 − 1 RowSpan ∈ Gr ≥ 0 (2 , 4) 0 2 1 1 u 1 u 2 u 3 u 4 u 1 ∆ 13 = 1 , ∆ 24 = 3 , ∆ 12 = 2 , ∆ 34 = 1 , ∆ 14 = 1 , ∆ 23 = 1 . In Gr(2 , 4), we have a Pl¨ ucker relation: ∆ 13 ∆ 24 = ∆ 12 ∆ 34 + ∆ 14 ∆ 23 . Top cell: ∆ 13 , ∆ 24 , ∆ 12 , ∆ 34 , ∆ 14 , ∆ 23 > 0 Codimension 1 cells: ∆ 12 = 0 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 14 / 29

  52. Example: Gr ≥ 0 (2 , 4) u 2 u 4 u 3 � 1 � 1 0 − 1 RowSpan ∈ Gr ≥ 0 (2 , 4) 0 2 1 1 u 1 u 2 u 3 u 4 u 1 ∆ 13 = 1 , ∆ 24 = 3 , ∆ 12 = 2 , ∆ 34 = 1 , ∆ 14 = 1 , ∆ 23 = 1 . In Gr(2 , 4), we have a Pl¨ ucker relation: ∆ 13 ∆ 24 = ∆ 12 ∆ 34 + ∆ 14 ∆ 23 . Top cell: ∆ 13 , ∆ 24 , ∆ 12 , ∆ 34 , ∆ 14 , ∆ 23 > 0 Codimension 1 cells: ∆ 12 = 0, ∆ 23 = 0 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 14 / 29

  53. Example: Gr ≥ 0 (2 , 4) u 2 u 4 u 3 � 1 � 1 0 − 1 RowSpan ∈ Gr ≥ 0 (2 , 4) 0 2 1 1 u 1 u 2 u 3 u 4 u 1 ∆ 13 = 1 , ∆ 24 = 3 , ∆ 12 = 2 , ∆ 34 = 1 , ∆ 14 = 1 , ∆ 23 = 1 . In Gr(2 , 4), we have a Pl¨ ucker relation: ∆ 13 ∆ 24 = ∆ 12 ∆ 34 + ∆ 14 ∆ 23 . Top cell: ∆ 13 , ∆ 24 , ∆ 12 , ∆ 34 , ∆ 14 , ∆ 23 > 0 Codimension 1 cells: ∆ 12 = 0, ∆ 23 = 0 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 14 / 29

  54. Example: Gr ≥ 0 (2 , 4) u 2 u 4 u 3 � 1 � 1 0 − 1 RowSpan ∈ Gr ≥ 0 (2 , 4) 0 2 1 1 u 1 u 2 u 3 u 4 u 1 ∆ 13 = 1 , ∆ 24 = 3 , ∆ 12 = 2 , ∆ 34 = 1 , ∆ 14 = 1 , ∆ 23 = 1 . In Gr(2 , 4), we have a Pl¨ ucker relation: ∆ 13 ∆ 24 = ∆ 12 ∆ 34 + ∆ 14 ∆ 23 . Top cell: ∆ 13 , ∆ 24 , ∆ 12 , ∆ 34 , ∆ 14 , ∆ 23 > 0 Codimension 1 cells: ∆ 12 = 0, ∆ 23 = 0, ∆ 34 = 0 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 14 / 29

  55. Example: Gr ≥ 0 (2 , 4) u 2 u 4 u 3 � 1 � 1 0 − 1 RowSpan ∈ Gr ≥ 0 (2 , 4) 0 2 1 1 u 1 u 2 u 3 u 4 u 1 ∆ 13 = 1 , ∆ 24 = 3 , ∆ 12 = 2 , ∆ 34 = 1 , ∆ 14 = 1 , ∆ 23 = 1 . In Gr(2 , 4), we have a Pl¨ ucker relation: ∆ 13 ∆ 24 = ∆ 12 ∆ 34 + ∆ 14 ∆ 23 . Top cell: ∆ 13 , ∆ 24 , ∆ 12 , ∆ 34 , ∆ 14 , ∆ 23 > 0 Codimension 1 cells: ∆ 12 = 0, ∆ 23 = 0, ∆ 34 = 0 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 14 / 29

  56. Example: Gr ≥ 0 (2 , 4) u 2 u 3 � 1 � 1 0 − 1 RowSpan ∈ Gr ≥ 0 (2 , 4) 0 2 1 1 u 4 u 1 u 2 u 3 u 4 u 1 ∆ 13 = 1 , ∆ 24 = 3 , ∆ 12 = 2 , ∆ 34 = 1 , ∆ 14 = 1 , ∆ 23 = 1 . In Gr(2 , 4), we have a Pl¨ ucker relation: ∆ 13 ∆ 24 = ∆ 12 ∆ 34 + ∆ 14 ∆ 23 . Top cell: ∆ 13 , ∆ 24 , ∆ 12 , ∆ 34 , ∆ 14 , ∆ 23 > 0 Codimension 1 cells: ∆ 12 = 0, ∆ 23 = 0, ∆ 34 = 0, ∆ 14 = 0. Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 14 / 29

  57. Example: Gr ≥ 0 (2 , 4) u 2 u 4 u 3 � 1 � 1 0 − 1 RowSpan ∈ Gr ≥ 0 (2 , 4) 0 2 1 1 u 1 u 2 u 3 u 4 u 1 ∆ 13 = 1 , ∆ 24 = 3 , ∆ 12 = 2 , ∆ 34 = 1 , ∆ 14 = 1 , ∆ 23 = 1 . In Gr(2 , 4), we have a Pl¨ ucker relation: ∆ 13 ∆ 24 = ∆ 12 ∆ 34 + ∆ 14 ∆ 23 . Top cell: ∆ 13 , ∆ 24 , ∆ 12 , ∆ 34 , ∆ 14 , ∆ 23 > 0 Codimension 1 cells: ∆ 12 = 0, ∆ 23 = 0, ∆ 34 = 0, ∆ 14 = 0. Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 14 / 29

  58. Example: Gr ≥ 0 (2 , 4) u 2 u 3 u 1 � 1 � 1 0 − 1 RowSpan ∈ Gr ≥ 0 (2 , 4) 0 2 1 1 u 1 u 2 u 3 u 4 u 4 ∆ 13 = 1 , ∆ 24 = 3 , ∆ 12 = 2 , ∆ 34 = 1 , ∆ 14 = 1 , ∆ 23 = 1 . In Gr(2 , 4), we have a Pl¨ ucker relation: ∆ 13 ∆ 24 = ∆ 12 ∆ 34 + ∆ 14 ∆ 23 . Top cell: ∆ 13 , ∆ 24 , ∆ 12 , ∆ 34 , ∆ 14 , ∆ 23 > 0 Codimension 1 cells: ∆ 12 = 0, ∆ 23 = 0, ∆ 34 = 0, ∆ 14 = 0. Codimension 2 cell: ∆ 12 = ∆ 14 = ∆ 24 = 0. Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 14 / 29

  59. The topology of Gr ≥ 0 ( k , n ) Theorem (Postnikov (2006)) Each boundary cell (some ∆ I > 0 and the rest ∆ J = 0 ) is an open ball. Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 15 / 29

  60. The topology of Gr ≥ 0 ( k , n ) Theorem (Postnikov (2006)) Each boundary cell (some ∆ I > 0 and the rest ∆ J = 0 ) is an open ball. Conjecture (Postnikov (2006)) The closure of each boundary cell is homeomorphic to a closed ball. Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 15 / 29

  61. The topology of Gr ≥ 0 ( k , n ) Theorem (Postnikov (2006)) Each boundary cell (some ∆ I > 0 and the rest ∆ J = 0 ) is an open ball. Conjecture (Postnikov (2006)) The closure of each boundary cell is homeomorphic to a closed ball. Williams (2007), Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 15 / 29

  62. The topology of Gr ≥ 0 ( k , n ) Theorem (Postnikov (2006)) Each boundary cell (some ∆ I > 0 and the rest ∆ J = 0 ) is an open ball. Conjecture (Postnikov (2006)) The closure of each boundary cell is homeomorphic to a closed ball. Williams (2007), Postnikov–Speyer–Williams (2009), Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 15 / 29

  63. The topology of Gr ≥ 0 ( k , n ) Theorem (Postnikov (2006)) Each boundary cell (some ∆ I > 0 and the rest ∆ J = 0 ) is an open ball. Conjecture (Postnikov (2006)) The closure of each boundary cell is homeomorphic to a closed ball. Williams (2007), Postnikov–Speyer–Williams (2009), Rietsch–Williams (2010). Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 15 / 29

  64. The topology of Gr ≥ 0 ( k , n ) Theorem (Postnikov (2006)) Each boundary cell (some ∆ I > 0 and the rest ∆ J = 0 ) is an open ball. Conjecture (Postnikov (2006)) The closure of each boundary cell is homeomorphic to a closed ball. Williams (2007), Postnikov–Speyer–Williams (2009), Rietsch–Williams (2010). Theorem (G.–Karp–Lam (2017)) Gr ≥ 0 ( k , n ) is homeomorphic to a k ( n − k ) -dimensional closed ball. Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 15 / 29

  65. The topology of Gr ≥ 0 ( k , n ) Theorem (Postnikov (2006)) Each boundary cell (some ∆ I > 0 and the rest ∆ J = 0 ) is an open ball. Conjecture (Postnikov (2006)) The closure of each boundary cell is homeomorphic to a closed ball. Williams (2007), Postnikov–Speyer–Williams (2009), Rietsch–Williams (2010). Theorem (G.–Karp–Lam (2017)) Gr ≥ 0 ( k , n ) is homeomorphic to a k ( n − k ) -dimensional closed ball. Theorem (G.–Karp–Lam (2019+)) The closure of each boundary cell is homeomorphic to a closed ball. Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 15 / 29

  66. The topology of Gr ≥ 0 ( k , n ) Theorem (Postnikov (2006)) Each boundary cell (some ∆ I > 0 and the rest ∆ J = 0 ) is an open ball. Conjecture (Postnikov (2006)) The closure of each boundary cell is homeomorphic to a closed ball. Williams (2007), Postnikov–Speyer–Williams (2009), Rietsch–Williams (2010). Theorem (G.–Karp–Lam (2017)) Gr ≥ 0 ( k , n ) is homeomorphic to a k ( n − k ) -dimensional closed ball. Theorem (G.–Karp–Lam (2019+)) The closure of each boundary cell is homeomorphic to a closed ball. Theorem (Smale (1960), Freedman (1982), Perelman (2003)) Let C be a compact contractible topological manifold whose boundary is homeomorphic to a sphere. Then C is homeomorphic to a closed ball. Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 15 / 29

  67. The totally nonnegative orthogonal Grassmannian N = 4 supersymmetric Gr ≥ 0 ( k , n ) ← → amplituhedron ← → Yang–Mills theory Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 16 / 29

  68. The totally nonnegative orthogonal Grassmannian N = 4 supersymmetric Gr ≥ 0 ( k , n ) ← → amplituhedron ← → Yang–Mills theory N = 6 supersymmetric ? OG ≥ 0 ( n , 2 n ) ← → ← → Chern-Simons matter theory Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 16 / 29

  69. The totally nonnegative orthogonal Grassmannian N = 4 supersymmetric Gr ≥ 0 ( k , n ) ← → amplituhedron ← → Yang–Mills theory N = 6 supersymmetric ? OG ≥ 0 ( n , 2 n ) ← → ← → Chern-Simons matter theory Recall: Gr ≥ 0 ( k , n ) := { W ∈ Gr( k , n ) | ∆ I ( W ) ≥ 0 for all I } . Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 16 / 29

  70. The totally nonnegative orthogonal Grassmannian N = 4 supersymmetric Gr ≥ 0 ( k , n ) ← → amplituhedron ← → Yang–Mills theory N = 6 supersymmetric ? OG ≥ 0 ( n , 2 n ) ← → ← → Chern-Simons matter theory Recall: Gr ≥ 0 ( k , n ) := { W ∈ Gr( k , n ) | ∆ I ( W ) ≥ 0 for all I } . The orthogonal Grassmannian: OG( n , 2 n ) := { W ∈ Gr( n , 2 n ) | ∆ I ( W ) = ∆ [2 n ] \ I ( W ) for all I } . Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 16 / 29

  71. The totally nonnegative orthogonal Grassmannian N = 4 supersymmetric Gr ≥ 0 ( k , n ) ← → amplituhedron ← → Yang–Mills theory N = 6 supersymmetric ? OG ≥ 0 ( n , 2 n ) ← → ← → Chern-Simons matter theory Recall: Gr ≥ 0 ( k , n ) := { W ∈ Gr( k , n ) | ∆ I ( W ) ≥ 0 for all I } . The orthogonal Grassmannian: OG( n , 2 n ) := { W ∈ Gr( n , 2 n ) | ∆ I ( W ) = ∆ [2 n ] \ I ( W ) for all I } . Definition (Huang–Wen (2013)) The totally nonnegative orthogonal Grassmannian: OG ≥ 0 ( n , 2 n ) := OG( n , 2 n ) ∩ Gr ≥ 0 ( n , 2 n ) Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 16 / 29

  72. The totally nonnegative orthogonal Grassmannian N = 4 supersymmetric Gr ≥ 0 ( k , n ) ← → amplituhedron ← → Yang–Mills theory N = 6 supersymmetric ? OG ≥ 0 ( n , 2 n ) ← → ← → Chern-Simons matter theory Recall: Gr ≥ 0 ( k , n ) := { W ∈ Gr( k , n ) | ∆ I ( W ) ≥ 0 for all I } . The orthogonal Grassmannian: OG( n , 2 n ) := { W ∈ Gr( n , 2 n ) | ∆ I ( W ) = ∆ [2 n ] \ I ( W ) for all I } . Definition (Huang–Wen (2013)) The totally nonnegative orthogonal Grassmannian: OG ≥ 0 ( n , 2 n ) := OG( n , 2 n ) ∩ Gr ≥ 0 ( n , 2 n ), i.e., OG ≥ 0 ( n , 2 n ) := { W ∈ Gr( n , 2 n ) | ∆ I ( W ) = ∆ [2 n ] \ I ( W ) ≥ 0 for all I } . Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 16 / 29

  73. The totally nonnegative orthogonal Grassmannian N = 4 supersymmetric Gr ≥ 0 ( k , n ) ← → amplituhedron ← → Yang–Mills theory N = 6 supersymmetric ? OG ≥ 0 ( n , 2 n ) ← → ← → Chern-Simons matter theory Recall: Gr ≥ 0 ( k , n ) := { W ∈ Gr( k , n ) | ∆ I ( W ) ≥ 0 for all I } . The orthogonal Grassmannian: OG( n , 2 n ) := { W ∈ Gr( n , 2 n ) | ∆ I ( W ) = ∆ [2 n ] \ I ( W ) for all I } . Definition (Huang–Wen (2013)) The totally nonnegative orthogonal Grassmannian: OG ≥ 0 ( n , 2 n ) := OG( n , 2 n ) ∩ Gr ≥ 0 ( n , 2 n ), i.e., OG ≥ 0 ( n , 2 n ) := { W ∈ Gr( n , 2 n ) | ∆ I ( W ) = ∆ [2 n ] \ I ( W ) ≥ 0 for all I } . dim(Gr ≥ 0 ( n , 2 n )) = n 2 Pavel Galashin (MIT) Ising model and total positivity U of Toronto, 01/07/2019 16 / 29

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