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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results Bounding entropies of hard squares and friends How to pick a good vector Andrew Rechnitzer Yao-ban Chan Melbourne, April 2013 Rechnitzer Packing bits 2d Bounds Upper

  1. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results B OUNDS Back to hardsquares.. . • No reason that κ should have a “nice” expression. • So try to find tight bounds. Most approaches based on transfer matrices Big problem — # states grows exponentially with width Rechnitzer

  2. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results T RANSFER MATRIX T w = column-to-column TM for hard squares in strip of width w Rechnitzer

  3. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results T RANSFER MATRIX T w = column-to-column TM for hard squares in strip of width w Rechnitzer

  4. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results T RANSFER MATRIX T w = column-to-column TM for hard squares in strip of width w w →∞ Λ 1 / w κ = lim where Λ w is dominant eigenvalue w Rechnitzer

  5. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results U SEFUL IDEAS FROM L INEAR A LGEBRA 101 Symmetric matrix V • Eigenvalues λ 1 , . . . , λ n all real Rechnitzer

  6. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results U SEFUL IDEAS FROM L INEAR A LGEBRA 101 Symmetric matrix V • Eigenvalues λ 1 , . . . , λ n all real • Min-max Theorem — for any non-trivial vector x , λ min ≤ � x | V | x � ≤ λ max � x | x � Rechnitzer

  7. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results U SEFUL IDEAS FROM L INEAR A LGEBRA 101 Symmetric matrix V • Eigenvalues λ 1 , . . . , λ n all real • Min-max Theorem — for any non-trivial vector x , λ min ≤ � x | V | x � ≤ λ max � x | x � • Trace of power Tr V k = λ k 1 + λ k 2 + · · · + λ k n Rechnitzer

  8. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results U SEFUL IDEAS FROM L INEAR A LGEBRA 101 Symmetric matrix V • Eigenvalues λ 1 , . . . , λ n all real • Min-max Theorem — for any non-trivial vector x , λ min ≤ � x | V | x � ≤ λ max � x | x � • Trace of power Tr V k = λ k 1 + λ k 2 + · · · + λ k n Tr V 2 k = λ 2 k 1 + λ 2 k 2 + · · · + λ 2 k n ≥ λ 2 k max Rechnitzer

  9. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results U SEFUL IDEAS FROM L INEAR A LGEBRA 101 Symmetric matrix V • Eigenvalues λ 1 , . . . , λ n all real • Min-max Theorem — for any non-trivial vector x , λ min ≤ � x | V | x � ≤ λ max � x | x � • Trace of power Tr V k = λ k 1 + λ k 2 + · · · + λ k n Tr V 2 k = λ 2 k 1 + λ 2 k 2 + · · · + λ 2 k n ≥ λ 2 k max Leverage these to get good bounds [Engel 1990] and [Calkin & Wilf 1998] Rechnitzer

  10. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results T RACE TRICK Rewrite trace Tr V 2 k = � V ψ 0 ,ψ 1 V ψ 1 ,ψ 2 . . . V ψ 2 k − 1 ,ψ 0 Sum is over all sequences of states, but only “legal” ones count Rechnitzer

  11. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results T RACE TRICK Rewrite trace Tr V 2 k = � V ψ 0 ,ψ 1 V ψ 1 ,ψ 2 . . . V ψ 2 k − 1 ,ψ 0 Sum is over all sequences of states, but only “legal” ones count Rechnitzer

  12. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results T RACE TRICK Rewrite trace Tr V 2 k = � V ψ 0 ,ψ 1 V ψ 1 ,ψ 2 . . . V ψ 2 k − 1 ,ψ 0 Sum is over all sequences of states, but only “legal” ones count So Tr T 2 k w is equivalent to “legal” configurations on rings � � � � Tr T 2 k � B w − 1 w = 1 � 1 � � 2 k Rechnitzer

  13. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results T RACE TRICK Rewrite trace Tr V 2 k = � V ψ 0 ,ψ 1 V ψ 1 ,ψ 2 . . . V ψ 2 k − 1 ,ψ 0 Sum is over all sequences of states, but only “legal” ones count So Tr T 2 k w is equivalent to “legal” configurations on rings � � � � Tr T 2 k � B w − 1 w = 1 � 1 � � 2 k Sneaky — “width” is now exponent. Rechnitzer

  14. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results L IMITS So build TM for rings B 2 k — also grows exponentially with circumference. Rechnitzer

  15. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results L IMITS So build TM for rings B 2 k — also grows exponentially with circumference. � � � � Λ 2 k w ≤ Tr T 2 k � B w − 1 w = 1 � 1 � � 2 k Rechnitzer

  16. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results L IMITS So build TM for rings B 2 k — also grows exponentially with circumference. � � � � Λ 2 k w ≤ Tr T 2 k � B w − 1 w = 1 � 1 � � 2 k Raise to 1 / w and let width → ∞ � 1 / w � 1 / w Λ 2 k / w � B w − 1 Tr T 2 k � � � 1 ≤ � = � 1 w w 2 k ↓ ↓ κ 2 k ≤ ξ 2 k Upper bound Let B 2 k be the TM for system on ring of circumference 2 k , then κ ≤ ξ 1 / 2 k 2 k where ξ 2 k is dominant eigenvalue of B 2 k . Rechnitzer

  17. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results R ESULTS ξ 2 = 2 . 41421356237309504 . . . κ ≤ 1 . 55377397403003730 . . . Rechnitzer

  18. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results R ESULTS ξ 2 = 2 . 41421356237309504 . . . κ ≤ 1 . 55377397403003730 . . . ξ 4 = 5 . 15632517465866169 . . . κ ≤ 1 . 50690222590181180 . . . Rechnitzer

  19. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results R ESULTS ξ 2 = 2 . 41421356237309504 . . . κ ≤ 1 . 55377397403003730 . . . ξ 4 = 5 . 15632517465866169 . . . κ ≤ 1 . 50690222590181180 . . . ξ 6 = 11 . 5517095660481450 . . . κ ≤ 1 . 50351480947590302 . . . [Calkin & Wilf 1998] Rechnitzer

  20. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results R ESULTS ξ 2 = 2 . 41421356237309504 . . . κ ≤ 1 . 55377397403003730 . . . ξ 4 = 5 . 15632517465866169 . . . κ ≤ 1 . 50690222590181180 . . . ξ 6 = 11 . 5517095660481450 . . . κ ≤ 1 . 50351480947590302 . . . [Calkin & Wilf 1998] ξ 36 = 2349759 . 74655388695 . . . κ ≤ 1 . 5030480824753399273 [Friedland, Lundow & Markstr¨ om 2010] Rechnitzer

  21. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results R ESULTS ξ 2 = 2 . 41421356237309504 . . . κ ≤ 1 . 55377397403003730 . . . ξ 4 = 5 . 15632517465866169 . . . κ ≤ 1 . 50690222590181180 . . . ξ 6 = 11 . 5517095660481450 . . . κ ≤ 1 . 50351480947590302 . . . [Calkin & Wilf 1998] ξ 36 = 2349759 . 74655388695 . . . κ ≤ 1 . 5030480824753399273 [Friedland, Lundow & Markstr¨ om 2010] Huge transfer matrix — use symmetries to compress it. Rechnitzer

  22. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results R AYLEIGH QUOTIENTS Min-max theorem λ min ≤ � x | V | x � ≤ λ max � x | x � Rechnitzer

  23. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results R AYLEIGH QUOTIENTS Min-max theorem λ min ≤ � x | V | x � ≤ λ max � x | x � So the simplest idea — set | x � = | 1 � . Rechnitzer

  24. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results R AYLEIGH QUOTIENTS Min-max theorem λ min ≤ � x | V | x � ≤ λ max � x | x � So the simplest idea — set | x � = | 1 � . Λ w ≥ � 1 | T w | 1 � � 1 | 1 � Rechnitzer

  25. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results R AYLEIGH QUOTIENTS Min-max theorem λ min ≤ � x | V | x � ≤ λ max � x | x � So the simplest idea — set | x � = | 1 � . Λ w ≥ � 1 | T w | 1 � � 1 | 1 � For fixed w this is silly — instead compute the eigenvalue by power method. Rechnitzer

  26. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results R AYLEIGH QUOTIENTS Min-max theorem λ min ≤ � x | V | x � ≤ λ max � x | x � So the simplest idea — set | x � = | 1 � . Λ w ≥ � 1 | T w | 1 � � 1 | 1 � For fixed w this is silly — instead compute the eigenvalue by power method. But if we can choose a better vector. . . Rechnitzer

  27. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results S NEAKY TRICKS AGAIN Vector | 1 � a poor choice. � T p � 1 � � � � 1 Λ p w w ≥ � 1 | 1 � Rechnitzer

  28. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results S NEAKY TRICKS AGAIN Power method — replace | 1 � with T q w | 1 � . T q � T p � T q � � � � w 1 w 1 w Λ p w ≥ T q � T q � � � w 1 w 1 Rechnitzer

  29. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results S NEAKY TRICKS AGAIN Massage denominator � � � � T q � T q � T 2 q � � � = w 1 w 1 1 � � � 1 w Rechnitzer

  30. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results S NEAKY TRICKS AGAIN Massage denominator � � � � T q � T q � T 2 q � � � = w 1 w 1 1 � � � 1 w Rechnitzer

  31. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results S NEAKY TRICKS AGAIN All configs in w × 2 q rectangle = configs in 2 q × w rectangle � � � � T q � T q � T 2 q � T w � 1 � � � � � � � = = w 1 w 1 1 � � 1 � 1 w 2 q Sneaky — width becomes exponent Rechnitzer

  32. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results S NEAKY TRICKS AGAIN Look at numerator now T q � T p � T q � � � � w 1 w 1 Λ p w w ≥ T q � T q � � � w 1 w 1 Rechnitzer

  33. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results S NEAKY TRICKS AGAIN Massage things a little � � � � T q � T p � T q � T 2 q + p � � � � = w 1 w 1 1 � � � 1 w w Rechnitzer

  34. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results S NEAKY TRICKS AGAIN Massage things a little � � � � T q � T p � T q � T 2 q + p � � � � = w 1 w 1 1 � � � 1 w w Rechnitzer

  35. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results S NEAKY TRICKS AGAIN Again use the x ↔ y symmetry � � � � T q � T p � T q � T 2 q + p � T w � 1 � � � � � � � � = = w 1 w 1 1 � � � 1 1 w w 2 q + p Sneaky — width becomes exponent Rechnitzer

  36. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results R ESULTS Putting this together � T w � � � 1 � � 1 Λ p 2 q + p w ≥ � � � � � T w 1 � � � 1 2 q Now raise to 1 / w and let w → ∞ Rechnitzer

  37. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results R ESULTS Putting this together � T w � � � � 1 � 1 Λ p 2 q + p w ≥ � � � � � T w 1 � � � 1 2 q Now raise to 1 / w and let w → ∞ Lower bound [Calkin & Wilf 1998] For any p , q ≥ 1 κ p ≥ Λ 2 q + p Λ 2 q Rechnitzer

  38. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results R ESULTS Putting this together � T w � � � 1 � � 1 Λ p 2 q + p w ≥ � � � � � T w 1 � � � 1 2 q Now raise to 1 / w and let w → ∞ Lower bound [Calkin & Wilf 1998] For any p , q ≥ 1 κ p ≥ Λ 2 q + p Λ 2 q • κ ≥ 1 . 50304768131466259 . . . ( p = 3 , q = 2 ) [Calkin & Wilf] • κ ≥ 1 . 50304808247533226 . . . ( p = 1 , q = 13 ) [Friedland et al] Rechnitzer

  39. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results P ICK A BETTER VECTOR • We use corner transfer matrix formalism to pick a better vector. Rechnitzer

  40. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results P ICK A BETTER VECTOR • We use corner transfer matrix formalism to pick a better vector. • Corner transfer matrices used to study lattice gas & magnet models [Baxter 1968] Rechnitzer

  41. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results P ICK A BETTER VECTOR • We use corner transfer matrix formalism to pick a better vector. • Corner transfer matrices used to study lattice gas & magnet models [Baxter 1968] • Very famously lead to solution of hard hexagons [Baxter 1980] Rechnitzer

  42. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results H OW TO BUILD A VECTOR Each entry of vector corresponds to a state along the cut Rechnitzer

  43. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results H OW TO BUILD A VECTOR Each entry of vector corresponds to a state along the cut Baxter’s Ansatz which extends [Kramers & Wannier 1941] Build Rayleigh quotient with vector ψ ψ ( σ 1 , σ 2 , . . . , σ w ) = Tr [ F ( σ 1 , σ 2 ) F ( σ 2 , σ 3 ) . . . F ( σ w , σ 1 )] For some matrices F ( a , b ) . Rechnitzer

  44. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results W HAT DOES THIS LOOK LIKE ? • Can think of F as a “literal” half-row transfer matrix. — but it can be almost any matrix. Rechnitzer

  45. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results W HAT DOES THIS LOOK LIKE ? • Can think of F as a “literal” half-row transfer matrix. — but it can be almost any matrix. • Trace makes it a cylinder — doesn’t change bound. Rechnitzer

  46. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results R AYLEIGH QUOTIENT → T RACES Rayleigh quotient � ψ | T | ψ � = Tr S w Λ w ≥ � ψ | T w | ψ � � ψ | ψ � = Tr R w � ψ | ψ � Rechnitzer

  47. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results R AYLEIGH QUOTIENT → T RACES Rayleigh quotient � ψ | T | ψ � = Tr S w Λ w ≥ � ψ | T w | ψ � � ψ | ψ � = Tr R w � ψ | ψ � Rechnitzer

  48. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results R AYLEIGH QUOTIENT → T RACES Rayleigh quotient � ψ | T | ψ � = Tr S w Λ w ≥ � ψ | T w | ψ � � ψ | ψ � = Tr R w � ψ | ψ � Where ω = 1 if face valid else ω = 0. Rechnitzer

  49. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results T O GET A BOUND Lower bound w →∞ Λ 1 / w κ = lim w Rechnitzer

  50. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results T O GET A BOUND Lower bound � Tr S w � 1 / w w →∞ Λ 1 / w κ = lim ≥ lim w Tr R w w →∞ Rechnitzer

  51. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results T O GET A BOUND Lower bound � Tr S w � 1 / w = η w →∞ Λ 1 / w κ = lim ≥ lim w Tr R w ξ w →∞ where ξ, η are dominant eigenvalues of R and S . Rechnitzer

  52. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results T O GET A BOUND Lower bound � Tr S w � 1 / w = η w →∞ Λ 1 / w κ = lim ≥ lim w Tr R w ξ w →∞ where ξ, η are dominant eigenvalues of R and S . 1 Pick matrices F — note dimension need not be related to w 2 Form matrices R and S 3 Compute dominant eigenvalues of ξ, η . Rechnitzer

  53. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results T O GET A BOUND Lower bound � Tr S w � 1 / w = η w →∞ Λ 1 / w κ = lim ≥ lim w Tr R w ξ w →∞ where ξ, η are dominant eigenvalues of R and S . 1 Pick matrices F — note dimension need not be related to w 2 Form matrices R and S 3 Compute dominant eigenvalues of ξ, η . But how do we pick F ? Rechnitzer

  54. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results T O GET A BOUND Lower bound � Tr S w � 1 / w = η w →∞ Λ 1 / w κ = lim ≥ lim w Tr R w ξ w →∞ where ξ, η are dominant eigenvalues of R and S . 1 Pick matrices F — note dimension need not be related to w 2 Form matrices R and S 3 Compute dominant eigenvalues of ξ, η . But how do we pick F ? And where are these infamous “corner transfer matrices”? Rechnitzer

  55. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results E IGENVECTORS �→ EIGENMATRICES (?) R | X � = ξ | X � S | Y � = η | Y � | X � , | Y � eigenvectors of R and S . Rechnitzer

  56. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results E IGENVECTORS �→ EIGENMATRICES (?) � R | X � = ξ | X � F ( a , b ) X ( b ) F ( b , a ) = ξ X ( a ) b � a � b � S | Y � = η | Y � ω F ( a , c ) Y ( c , d ) F ( d , b ) = η Y ( a , b ) c d c , d X ( a ) , Y ( a , b ) ≈ “half-plane transfer matrices” Rechnitzer

  57. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results T O MAXIMISE , PLANES �→ CORNER × CORNER Baxter showed that Rayleigh quotient stationary when X ( a ) = A ( a ) 2 Y ( a , b ) = A ( a ) F ( a , b ) A ( b ) where A is half of X — a “corner transfer matrix” Baxter then carefully picked F to make things work. Rechnitzer

  58. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results R ENORMALISE INSTEAD • We have used “corner transfer matrix renormalisation group method” [Nishino & Okunishi 1996] • Related to density matrix renormalisation group method [White 1992] Rechnitzer

  59. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results R ENORMALISE INSTEAD • We have used “corner transfer matrix renormalisation group method” [Nishino & Okunishi 1996] • Related to density matrix renormalisation group method [White 1992] • The central idea = only keep important parts of A . Rechnitzer

  60. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results B UILD RECURSIVELY Start by building “literal” matrices. Let • A be corner transfer matrix for a 2 × 2 grid • F be the half-row / half-column transfer matrix for a 1 × 2 grid Rechnitzer

  61. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results B UILD RECURSIVELY Start by building “literal” matrices. Let • A be corner transfer matrix for a 2 × 2 grid • F be the half-row / half-column transfer matrix for a 1 × 2 grid • Then build larger matrices by � a � b � A l ( c ) | d , a = ω F ( c , d ) A ( b ) F ( b , a ) c d d � a � b F ( c , d ) | b , a = ω F ( b , a ) c d Rechnitzer

  62. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results B UILD RECURSIVELY Start by building “literal” matrices. Let • A be corner transfer matrix for a 2 × 2 grid • F be the half-row / half-column transfer matrix for a 1 × 2 grid • Then build larger matrices by � a � b � A l ( c ) | d , a = ω F ( c , d ) A ( b ) F ( b , a ) c d d � a � b F ( c , d ) | b , a = ω F ( b , a ) c d • Iterate until A and F are huge — they are still “literal”. Rechnitzer

  63. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results N OW ESTIMATE EIGENVALUES ξ, η • Look at eigenvalue equation: � � ξ X ( a ) = F ( a , b ) X ( b ) F ( b , a ) a a , b Rechnitzer

  64. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results N OW ESTIMATE EIGENVALUES ξ, η • Look at eigenvalue equation: A ( a ) 2 = � � F ( a , b ) A ( b ) 2 F ( b , a ) ξ a a , b Rechnitzer

  65. Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results N OW ESTIMATE EIGENVALUES ξ, η • Look at eigenvalue equation: A ( a ) 4 = � � A ( a ) F ( a , b ) A ( b ) 2 F ( b , a ) A ( a ) ξ a a , b Rechnitzer

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