Continuum Branching Observable in Higher Genus (Based on Discussion - PowerPoint PPT Presentation

Continuum Branching Observable in Higher Genus (Based on Discussion with Nicolai Reshetikhin) Matthew Bernard mattb@berkeley.edu Advanced Computational Biology Center , Berkeley Supported by The Lynn Bit Foundation , State of California

Abstract 2 For all fjxed suffjciently large genus g =0 , 1 , ⩾ 2 , multiedge connected , dual graph , we give a uniform bipartite observable of Grassmann kernel transfer matrices . On special hexagonal domain , we prove discriminant steepest descent of Grassmann kernel logarithmic asymptotics , and free Dirac Fermion convergence Ψ 12 × (1+ O (1)) . We conjecture : In large deviation functional , the Green’s function G for Dirichlet problem of variational principle minimizer is observable in the kernel asymptotics . Keywords: Continuum-branching , higher-genus , observable

1 Characterizations for triangular grids 3 • ◦ Bipartite implies no adjacent-black (-white) vertices for all V X = V X ⊔ V X : � � � � ξ ∩ D : � • i • ξ � = η ( i • ξ , i • η , D )= ∅ ξ = { i ξ , j ξ } : i � = j, | i ξ ∩ D | =1 V X = ; D = . Instance . 1 6 12 18 1 2 3 4 2 7 13 19 24 3 8 14 20 25 5 6 7 8 4 9 15 21 26 5 10 16 22 27 9 10 11 12 . 11 17 23 28 Non-instance . 1 2 3 4 � no bipartite structure � . 5 6 7 8 9 10 11 12

4 Derivation. R n r . v .X =( X i ξ j ξ ; i ξ , j ξ ∈ V X ) is Gaussian G ifg X = µ a . s ., for � � � � it ′ µ − 1 � E [exp( it ′ X )] = exp 2 t ′ Σ t � µ ξ = E [ X i ξ j ξ ] , Σ ξη = cov [ X i ξ j ξ , X i η j η ] � − ( X − µ ) T Σ − 1 ( X − µ ) � 1 ⇐ ⇒ P { X ∈ dx } = � exp dx (2 π ) n det Σ 2 ifg t ′ X = � ξ t ξ X i ξ j ξ is R - G, ∀ t ∈ R n , where X is R - G ifg X = µ a . s ., for � � � � i µ t − Σ t 2 � � µ = E [ X ] , Σ = var [ X ] , ∀ t ∈ R E [exp( itX )] = exp 2 � � − ( X − µ ) 2 1 ⇐ ⇒ P { X ∈ dx } = √ exp dx 2Σ 2 π Σ ⇒ X is R n - G ; ifg X i ξ j ξ ∈ X is independent ( Σ ξ , η � = ξ = 0 ) ⇐ where X is absolutely continuous ifg Σ is non-singular .

Derivation. ifg a.s. 5 √− 1 H , U T U = UU T = I ; = X ifg centered X ∈ R n , Hermitian H | U = e UX d � � is uniformly distributed on S n − 1 = X X x ∈ R n : � x � =1 � X � = � n � X 2 i ξ j ξ ξ =1 where X is standard ifg centered ( µ =0 ) and Σ= I | X i ξ j ξ ∼ N (0 , 1) . Derivation. For n centered Gaussian , resp . Maxwell , particle velocity X, � � � � � 2Σ t 2 � n � − 1 � t 2 1 + · · · + t 2 E [ itX ] = Φ( t ξ ) = Φ � Φ( t ) = exp n ξ =1 � � � � � 2Σ t 2 � 3 � − 1 � t 2 1 + t 2 2 + t 2 resp . E [ itX ] = Φ( t ξ ) = Φ � Φ( t ) = exp , Σ ⩾ 0 . 3 ξ =1 Remark. Taking n − → + ∞ , for support S, continuous density f ( x ): − 1 n log f ⊗ n ( X i 1 j 1 , . . . , X i n j n ) � f ( x ) log f ( x ) = 1 − → E [ − log f ( X )] = − 2 log(2 πe Σ) . S

1.1 is Partition 6 � � i ξ ∩ D : � Defjnition. An embedding X ⊂M g | V ξ � = η ( i ξ , i η , D )= ∅ X = � � partition σ ∈ Aut ( D ) ifg perfect-matching D = ξ = { i ξ , j ξ } : i � = j, | i ξ ∩ D | =1 ; D = { D, ∀ ξ } , M g orientable compact , X closed , g ≫ . 1 6 2 5 7 8 3 4 That is , σ implies � � 1 if ξ ∩ D � � � σ }| ) − 1 | Aut ( D ) | · ( |{ � 1 � � ∂D = V � = � σ i ξ j ξ = σ i ξ j ξ � � = n ln 2 + � n − 1 X 2 0 otherwise exp k =1 ln k ξ σ = σ | σ 2 ξ > σ 2 ξ − 1 ; ξ , n ∈ N + X =( i ξ ∩ D : | i ξ | =2 n ); σ =( σ 1 , . . . , σ 2 n ) , � V where X ⊂ M g is CW cell-complex i.e . face ≈ topological disk i.e . no hole .

7 2 )) ∼ = [ σ ] ∼ Derivation. (i) { [ σ ] } ∼ 2 ) ( Aut ( D ) / ( S n ×S n = { � σ } = ( S n ×S n � (2 n )! · 2 − ((1 / ε ) mod c ( X )) · e ln ( a ( X ) · b ( X )) � 1 / (2 | D | ) σ }| 1 / | D | ⩽ (ii) |{ � � � a, b, c ∈ R + ; n ⩾ 2; n ! · a ( X ) · b ( X ) � where min(deg( X )) ⩾ ⌊ 2 n − 3 ⌋ !! = � ⌊ n ⌋− 2 X i ξ j ξ ≡ X σ 2 ξ − 1 σ 2 ξ 2 k +1 k =0 S n ∼ = { ( σ 1 , . . . , σ 2 n ) , . . . , ( σ 2 n − 1 , σ 2 n , . . . , σ 1 , σ 2 ) } ; σ ∈ S 2 n − → Im ( Aut ( D )) � 2 ∼ � S n = { ( σ 1 , . . . , σ 2 n ) , . . . , ( σ 2 , σ 1 , . . . , σ 2 n , σ 2 n − 1 ) } ; [ σ ]= { σ Aut ( D ) } � (2 n )! � (iii) (2 n )(2 n − 1) = 2 = | E X | holds for complete graph X = K 2 n n !2 n 2 on objects : 1 2 3 4 6 11 30 1 14 15 5 6 7 8 66 5 10 19 29 2 13 16 9 10 11 12 • square grid domains . 4 9 • (regular) hexagonal grid domains .

8 def which equals By E [ σ i ξ j ξ σ i η j η ] = E [ σ i ξ j ξ ] ifg ξ = η , resp . zero if � ξ � = η ( i ξ , i η , D ) � = ∅ i.e . dimers of D sharing vertex : The local observable is dimer-dimer correlation i.e ., for (Boltzmann) weights ω η , the conditional probability � � � � k k � � σ i η j η σ i η j η = Prob ( i 1 ∩ D 1 , j 1 ∩ D 1 , . . . , i k ∩ D k , j k ∩ D k ) = E = η =1 η =1 � � � k σ i η j η ω ξ k � � = 1 D η η =1 ξ ∩ D η σ i η j η × Prob ( D η ) = � � Z × Z ( η | η =1 ,...,k ) ω ξ D η η =1 D ξ ∩ D � � Ξ ξ K T = e − Ξ D � � e − ⇒ 1 � ω D = ω ξ = K T ω D η � = 0 ⇐ � � Z ξ ∩ D ξ ∩ D � D η ∩ ( i 1 � = i 2 ,...i k ,j 1 ,...,j k ) � � � � � � � Ξ D = Ξ ξ , Z def = = ω ξ � = Prob ( D ) ⇐ ⇒ D = { i η , D η } D ξ ∩ D ξ ∩ D η D η for strict-sense positive partition function Z on dimer energy → R + | ( i ξ ∩ D, j ξ ∩ D ) �− X − → Ξ ξ . Ξ : E

9 def Defjnition. The space H X of height function h D or h is the whole of Z : = { { { h D : F Z } } | D ← } H X = X − → Z Z − � Bipartite surfaces � h ( F i − 1 ) + 1 / 3 if i • ξ is left on crossing ξ � ∩ D h ( F i ) = ξ is left on crossing ξ � ∩ D ; h ( F 0 ) = 0 . i − 1 ) − 1 / 3 if i ◦ h ( F Derivation. H X is given on the bipartite hexagonal X ⊂ R 2 by : 9 5 9 5 9 5 h + 2 h + 1 h + 1 3 3 3 h + 1 h + 1 h + 2 h + 2 3 , h h , 13 10 13 10 13 10 3 3 3 h + 2 h h 3 16 14 16 14 16 14 for any perfect-matching D ∈ D , and base-face normalization h D ( F 0 )=0 . Theorem. For all H X : (i) h D = h i.e . h X is path ( T ∗ ) and D independent . � � � (ii) Curl sum d X = d F = ω i ξ j ξ = 0 ifg F X is all co-cycles . F F ξ : ξ ∩ ∂ F

10 Proof Follows by divergence-free notion on X, that is , d ∗ i D 1 D 2 = d ∗ i D 1 − d ∗ i D 2 = 0 ifg F X is all co-cycles ; � +1 if i : i • ξ ∩ D � � � � fmow ω i ξ j ξ = − ω j ξ i ξ = d ∗ i D = d ∗ − 1 if i : i ◦ ω i ξ j ξ i = ξ ∩ D 0 otherwise . j ξ Defjnition. Skew plane partition is sequence { λ | λ ⊃ µ } of diagonal slices � � λ ( t ) = π i, i + t ∈ N | i ⩾ max(0 , − t ) , ∀ t ∈ Z for fjnite monotone ( π ij ⩾ π i + r, j + s ; r, s ⩾ 0 ) array in the generalized 3 D partition array π =( π ij : ( i, j ) ∈ N 2 | π ij =0 , ∀ i + j ≫ 0) of X ∗ cubes π ij . Remark. π is uniquely determined by X ∗ bijection (projection) map R 3 �− → R 2 ⊃ { ( t, h ) } : t = y − x, h = z − ( y + x ) / 2 , ∀ ( x, y, z ) ∈ R 3 for all cubes mod Z 3 ⩾ 0 projection , with boundary (base) condition (0 , 0 , 0) . The centers of the horizontal hexagonal tiling is given by . �� �� ⊂ Z × 1 π C = i − j, π ij − ( i + j − 1) / 2 2 Z .

11 Cubes : 2 D mixing algorithm 27 28 6 11 43 24 69 30 1 31 14 15 44 72 78 103 66 36 5 10 35 19 26 48 83 136 108 57 29 2 32 13 16 45 71 79 90 137 100 65 37 4 9 34 18 25 47 84 102 113 59 20 3 33 12 17 46 70 80 91 138 105 64 39 7 8 42 23 68 74 85 139 60 21 41 22 62 51 76 81 123 125 52 126 127