Excursion into the Mayer expansion Particles at x 1 , . . . , x n in a finite set Λ. Grand canonical partition function ∞ z n � � � Z = (1 − f ij ) , . n ! n =0 x ∈ Λ n 1 ≤ i < j ≤ n = ✶ { xi , xj incompatible } By expanding the product Z becomes a sum over all graphs – connected and disconnected.
Excursion into the Mayer expansion Particles at x 1 , . . . , x n in a finite set Λ. Grand canonical partition function ∞ z n � � � Z = (1 − f ij ) , . n ! n =0 x ∈ Λ n 1 ≤ i < j ≤ n = ✶ { xi , xj incompatible } By expanding the product Z becomes a sum over all graphs – connected and disconnected. ∞ z n � � � � Mayer’s first theorem: log Z ∼ ( − f ij ) n ! n =1 G ∈C ( n ) x ∈ Λ n ij ∈ G
Excursion into the Mayer expansion Particles at x 1 , . . . , x n in a finite set Λ. Grand canonical partition function ∞ z n � � � Z = (1 − f ij ) , . n ! n =0 x ∈ Λ n 1 ≤ i < j ≤ n = ✶ { xi , xj incompatible } By expanding the product Z becomes a sum over all graphs – connected and disconnected. ∞ z n � � � � Mayer’s first theorem: log Z ∼ ( − f ij ) n ! n =1 G ∈C ( n ) x ∈ Λ n ij ∈ G { connected graphs with vertices 1 , . . . , n }
Excursion into the Mayer expansion Particles at x 1 , . . . , x n in a finite set Λ. Grand canonical partition function ∞ z n � � � Z = (1 − f ij ) , . n ! n =0 x ∈ Λ n 1 ≤ i < j ≤ n = ✶ { xi , xj incompatible } By expanding the product Z becomes a sum over all graphs – connected and disconnected. ∞ z n � � � � Mayer’s first theorem: log Z ∼ ( − f ij ) n ! n =1 G ∈C ( n ) x ∈ Λ n ij ∈ G { connected graphs with vertices 1 , . . . , n } Penrose reduced the sum over C ( n ) to a sum over the set T ( n ) of tree graphs = minimally connected graphs.
For each n choose an order on all possible edges
For each n choose an order on all possible edges Complete graph on n = 5 vertices
For each n choose an order on all possible edges Complete graph on n = 5 vertices
For each n choose an order on all possible edges Complete graph on n = 5 vertices 8 2 5 3 9 1 4 6 10 7
For each n choose an order on all possible edges Complete graph on n = 5 vertices 8 2 5 3 9 1 4 Orders edges for n = 5 6 10 7
Define Kruskal map k : C ( n ) �→ T ( n )
Define Kruskal map k : C ( n ) �→ T ( n ) For G in C ( n ) pick edges in order discarding those that form a loop.
Define Kruskal map k : C ( n ) �→ T ( n ) For G in C ( n ) pick edges in order discarding those that form a loop. 8 5 1 4 6 7
Define Kruskal map k : C ( n ) �→ T ( n ) For G in C ( n ) pick edges in order discarding those that form a loop. 8 5 1 4 6 7
Define Kruskal map k : C ( n ) �→ T ( n ) For G in C ( n ) pick edges in order discarding those that form a loop. 8 5 1 4 6 7
Define Kruskal map k : C ( n ) �→ T ( n ) For G in C ( n ) pick edges in order discarding those that form a loop. 8 5 1 4 6 7
Define Kruskal map k : C ( n ) �→ T ( n ) For G in C ( n ) pick edges in order discarding those that form a loop. 8 5 1 4 6 7
Define Kruskal map k : C ( n ) �→ T ( n ) For G in C ( n ) pick edges in order discarding those that form a loop. 8 5 1 4 6 7 connected graph G is mapped to tree subgraph T
The maximal graph M ( T ) By construction, for any tree, k ( T ) = T .
The maximal graph M ( T ) By construction, for any tree, k ( T ) = T . Given a tree T , add all edges such that the resulting graph M is still mapped by k to T . One can add edges in any order to reach the same M .
The maximal graph M ( T ) By construction, for any tree, k ( T ) = T . Given a tree T , add all edges such that the resulting graph M is still mapped by k to T . One can add edges in any order to reach the same M . All graphs G such that k ( G ) = T satisfy T ⊂ G ⊂ M .
The maximal graph M ( T ) By construction, for any tree, k ( T ) = T . Given a tree T , add all edges such that the resulting graph M is still mapped by k to T . One can add edges in any order to reach the same M . All graphs G such that k ( G ) = T satisfy T ⊂ G ⊂ M . Thus M = M ( T ) is the maximal graph such that k ( M ) = T .
The maximal graph M = M ( T ) such that k ( M ) = T
The maximal graph M = M ( T ) such that k ( M ) = T 8 2 5 3 9 1 4 6 10 7
The maximal graph M = M ( T ) such that k ( M ) = T 8 2 5 3 9 1 4 6 10 7 M is the red and dotted edges, i.e., all edges except the yellow edges.
The maximal graph M = M ( T ) such that k ( M ) = T 8 2 5 3 9 1 4 6 10 7 M is the red and dotted edges, i.e., all edges except the yellow edges. No graph with yellow edge 3 < max { 7 , 4 } can map to the red tree.
The maximal graph M = M ( T ) such that k ( M ) = T 8 2 5 3 9 1 4 6 10 7 M is the red and dotted edges, i.e., all edges except the yellow edges. No graph with yellow edge 3 < max { 7 , 4 } can map to the red tree. Likewise yellow edge 2 < max { 5 , 1 , 7 , 4 } .
The maximal graph M = M ( T ) such that k ( M ) = T 8 2 5 3 9 1 4 6 10 7 M is the red and dotted edges, i.e., all edges except the yellow edges. No graph with yellow edge 3 < max { 7 , 4 } can map to the red tree. Likewise yellow edge 2 < max { 5 , 1 , 7 , 4 } . The graphs that map to T are precisely graphs that contain T and any subset of the dotted lines.
Penrose resummation formula Lemma: ( − f ) G = ( − f ) T (1 − f ) M ( T ) \ T . � � G ∈C ( n ) T
Penrose resummation formula Lemma: � = ( − f ij ) ij ∈ G ( − f ) G = ( − f ) T (1 − f ) M ( T ) \ T . � � G ∈C ( n ) T
Penrose resummation formula Lemma: � = ( − f ij ) ij ∈ G ( − f ) G = ( − f ) T (1 − f ) M ( T ) \ T . � � G ∈C ( n ) T ∈ [0 , 1] if f ij ∈ [0 , 1]
Penrose resummation formula Lemma: � = ( − f ij ) ij ∈ G ( − f ) G = ( − f ) T (1 − f ) M ( T ) \ T . � � G ∈C ( n ) T ∈ [0 , 1] if f ij ∈ [0 , 1] This reduction from C ( n ) to T ( n ) easily implies that the expansion for log Z is absolutely convergent for z small.
Back to WSAW: define Greens function ∞ � z n � � Let G λ, z ( x ) := (1 − λ ✶ ω i = ω j ). n =0 ω ∈ Ω n ( x ) 1 ≤ i < j ≤ n
Back to WSAW: define Greens function ∞ � z n � � Let G λ, z ( x ) := (1 − λ ✶ ω i = ω j ). n =0 ω ∈ Ω n ( x ) 1 ≤ i < j ≤ n x ∈ Z d
Back to WSAW: define Greens function new parameter z ≥ 0 ∞ � z n � � Let G λ, z ( x ) := (1 − λ ✶ ω i = ω j ). n =0 ω ∈ Ω n ( x ) 1 ≤ i < j ≤ n x ∈ Z d
Back to WSAW: define Greens function new parameter z ≥ 0 ∞ � z n � � Let G λ, z ( x ) := (1 − λ ✶ ω i = ω j ). n =0 ω ∈ Ω n ( x ) 1 ≤ i < j ≤ n x ∈ Z d set of simple walks with n steps and ω n = x
Back to WSAW: define Greens function new parameter z ≥ 0 ∞ � z n � � Let G λ, z ( x ) := (1 − λ ✶ ω i = ω j ). n =0 ω ∈ Ω n ( x ) 1 ≤ i < j ≤ n x ∈ Z d set of simple walks with n steps and ω n = x � χ λ, z := G λ, z ( x ) is called the susceptibility. x ∈ Z d
Back to WSAW: define Greens function new parameter z ≥ 0 ∞ � z n � � Let G λ, z ( x ) := (1 − λ ✶ ω i = ω j ). n =0 ω ∈ Ω n ( x ) 1 ≤ i < j ≤ n x ∈ Z d set of simple walks with n steps and ω n = x � χ λ, z := G λ, z ( x ) is called the susceptibility. x ∈ Z d Let z c = z c ( λ ) be the radius of convergence of χ λ, z .
Back to WSAW: define Greens function new parameter z ≥ 0 ∞ � z n � � Let G λ, z ( x ) := (1 − λ ✶ ω i = ω j ). n =0 ω ∈ Ω n ( x ) 1 ≤ i < j ≤ n x ∈ Z d set of simple walks with n steps and ω n = x � χ λ, z := G λ, z ( x ) is called the susceptibility. x ∈ Z d Let z c = z c ( λ ) be the radius of convergence of χ λ, z . Objective: for d ≥ 5, λ small, G λ, z c ( λ ) ( x ) ≤ 2 G 0 , z c (0) ( x )
Comment This is called an infrared bound. Once we get it from the lace expansion other results such as (D). As n → ∞ , E λ, n | ω n | 2 ∼ cn for some c . are standard.
✶ Graphical expansion for G λ, z ( x ) In the formula for G λ, z ( x ) insert � � � (1 − f ij ) = ( − f ij ) 0 ≤ i < j ≤ n ij ∈ G G ∈G [0 ,..., n ]
Graphical expansion for G λ, z ( x ) In the formula for G λ, z ( x ) insert f ij = λ ✶ { ω i = ω j } � � � (1 − f ij ) = ( − f ij ) 0 ≤ i < j ≤ n ij ∈ G G ∈G [0 ,..., n ]
Graphical expansion for G λ, z ( x ) In the formula for G λ, z ( x ) insert f ij = λ ✶ { ω i = ω j } � � � (1 − f ij ) = ( − f ij ) 0 ≤ i < j ≤ n ij ∈ G G ∈G [0 ,..., n ] { graphs on vertices 0 , . . . , n }
Graphical expansion for G λ, z ( x ) In the formula for G λ, z ( x ) insert f ij = λ ✶ { ω i = ω j } � � � (1 − f ij ) = ( − f ij ) 0 ≤ i < j ≤ n ij ∈ G G ∈G [0 ,..., n ] { graphs on vertices 0 , . . . , n } f 26 0 1 2 6 n
Graphical expansion for G λ, z ( x ) In the formula for G λ, z ( x ) insert f ij = λ ✶ { ω i = ω j } � � � (1 − f ij ) = ( − f ij ) 0 ≤ i < j ≤ n ij ∈ G G ∈G [0 ,..., n ] { graphs on vertices 0 , . . . , n } f 26 0 1 2 6 n Definition: Markovian vertices have no arches over them
Graphical expansion for G λ, z ( x ) In the formula for G λ, z ( x ) insert f ij = λ ✶ { ω i = ω j } � � � (1 − f ij ) = ( − f ij ) 0 ≤ i < j ≤ n ij ∈ G G ∈G [0 ,..., n ] { graphs on vertices 0 , . . . , n } f 26 0 1 2 6 n Definition: Markovian vertices have no arches over them Say G ∈ C ( n ) if G has no Markovian points except 0.
Define Π λ, z ( x ) ∞ � z n � � � Π λ, z ( x ) := ( − λ ✶ ω i = ω j ) n =0 ω ∈ Ω n ( x ) G ∈C ( n ) ij ∈ G which is an expansion in graphs without Markovian points whereas ∞ � z n � � � G λ, z ( x ) = ( − λ ✶ ω i = ω j ) n =0 ij ∈ G ω ∈ Ω n ( x ) G ∈G ( n ) is an expansion in all possible graphs.
✶ Define k : C ( n ) → L ( n )
✶ Define k : C ( n ) → L ( n ) 0 n
✶ Define k : C ( n ) → L ( n ) 0 0 n n
✶ Define k : C ( n ) → L ( n ) 0 0 0 n n n
✶ Define k : C ( n ) → L ( n ) 0 0 0 0 n n n n
✶ Define k : C ( n ) → L ( n ) 0 0 0 0 0 n n n n n
✶ Define k : C ( n ) → L ( n ) 0 0 0 0 0 n n n n n Define L ( n ): L ∈ L ( n ) if L ∈ C ( n ) and is minimal.
✶ Define k : C ( n ) → L ( n ) 0 0 0 0 0 n n n n n Define L ( n ): L ∈ L ( n ) if L ∈ C ( n ) and is minimal. Lemma: This map k : C ( n ) → L ( n ) is such that
Define k : C ( n ) → L ( n ) 0 0 0 0 0 n n n n n Define L ( n ): L ∈ L ( n ) if L ∈ C ( n ) and is minimal. Lemma: This map k : C ( n ) → L ( n ) is such that ( − f ) G = ( − f ) L (1 − f ) M ( T ) \ L , � � f ij = λ ✶ ω i = ω j . G ∈C ( n ) L ∈L ( n )
✶ Instead of (1 − f ) M ( T ) \ L ≤ 1 used in Mayer
Instead of (1 − f ) M ( T ) \ L ≤ 1 used in Mayer 0 ≤ (1 − f ) M ( T ) \ L ≤ � (1 − λ ✶ ω i = ω j ) ij ∈ dotted edges
Instead of (1 − f ) M ( T ) \ L ≤ 1 used in Mayer 0 ≤ (1 − f ) M ( T ) \ L ≤ � (1 − λ ✶ ω i = ω j ) ij ∈ dotted edges enabling a bootstrap. G λ, z is expressed in terms of Π λ, z and Π λ, z is bounded in terms of G . A poor estimate on G can improve when passed through this circle.
Π bounded by G From the last Lemma and the definition of Π ∞ ( − f ) L (1 − f ) M ( T ) \ L , � � � z n Π λ, z ( x ) = n =0 ω ∈ Ω n ( x ) L ∈L ( n )
Π bounded by G From the last Lemma and the definition of Π ∞ ( − f ) L (1 − f ) M ( T ) \ L , � � � z n Π λ, z ( x ) = n =0 ω ∈ Ω n ( x ) L ∈L ( n ) From the (1 − f ) M ( T ) \ L inequality, | Π λ, z ( x ) | ≤ x + + + + where in the Feynman diagrams on the RHS each line is G λ, z ( · ) and each vertex has weight λ .
Schwinger-Dyson replaces log ↔ connected graphs For z ≤ z c , and if Π λ, z ∈ ℓ 1 , G λ ( z ) = G 0 ( z ) + G 0 ( z ) ∗ Π λ ( z ) ∗ G λ ( z )
Bootstrap See Lace Expansion for Dummies, Bolthausen–van der Hofstad–Kozma 2017. If d ≥ 5, λ small and z < z c ( λ ) the estimate G λ, z ( x ) ≤ 3 G 0 , z c (0) ( x ) passed through the bootstrap G λ, z → G λ, z → G λ, z implies the estimate G λ, z ( x ) ≤ 2 G 0 , z c (0) ( x ) (3 ⇒ 2) For z ≪ z c ( λ ), G λ, z ( x ) ≤ 2 G 0 , z c (0) ( x ) holds. continuity properties in z imply it holds z ≤ z c ( λ ).
✶ ✶ Percolation Mean-Field Critical Behaviour for Percolation in High Dimensions: (Hara–Slade 1990).
Percolation Mean-Field Critical Behaviour for Percolation in High Dimensions: (Hara–Slade 1990). � Whenever we expand and resum (1 − ✶ ω i = ω j ) we are 1 ≤ i < j ≤ n developing an inclusion-exclusion formula and the percolation lace expansion is an inclusion-exclusion formula modeled on the SAW expansion. The BK inequality plays enough of the role of 1 − ✶ ω ( s )= ω ( t ) ≤ 1 that one can get the analogue of x + + + +
✶ Spin models
✶ Spin models Lace expansion for the Ising model (high dimensions or finite range coupling). (A Sakai 2007)
✶ Spin models Lace expansion for the Ising model (high dimensions or finite range coupling). (A Sakai 2007) Application of the lace expansion to the ϕ 4 model (A. Sakai 2015).
Spin models Lace expansion for the Ising model (high dimensions or finite range coupling). (A Sakai 2007) Application of the lace expansion to the ϕ 4 model (A. Sakai 2015). In preparation: similar results as Akira Sakai, but also for the two component ϕ 4 model. (Brydges-Helmuth-Holmes) Correlation inequalities play the role of 1 − ✶ ω ( s )= ω ( t ) ≤ 1
Convergence of critical oriented percolation to super-Brownian motion above 4+1 dimensions. (van der Hofstad–Slade 2003).
Convergence of critical oriented percolation to super-Brownian motion above 4+1 dimensions. (van der Hofstad–Slade 2003). Incipient infinite cluster for spread-out oriented percolation above 4 + 1 dimensions (van der Hofstad–den Hollander–Slade 2002)
Convergence of critical oriented percolation to super-Brownian motion above 4+1 dimensions. (van der Hofstad–Slade 2003). Incipient infinite cluster for spread-out oriented percolation above 4 + 1 dimensions (van der Hofstad–den Hollander–Slade 2002) Existence, properties of the incipient infinite cluster for high-dimensional unoriented percolation V der Hofstad–J´ arai 2004.
Convergence of critical oriented percolation to super-Brownian motion above 4+1 dimensions. (van der Hofstad–Slade 2003). Incipient infinite cluster for spread-out oriented percolation above 4 + 1 dimensions (van der Hofstad–den Hollander–Slade 2002) Existence, properties of the incipient infinite cluster for high-dimensional unoriented percolation V der Hofstad–J´ arai 2004. Random Subgraphs Of Finite Graphs: The Phase Transition For The n-Cube. (Borgs–J. Chayes–v. der Hofstad-Slade–J. Spencer 2006)
Convergence of critical oriented percolation to super-Brownian motion above 4+1 dimensions. (van der Hofstad–Slade 2003). Incipient infinite cluster for spread-out oriented percolation above 4 + 1 dimensions (van der Hofstad–den Hollander–Slade 2002) Existence, properties of the incipient infinite cluster for high-dimensional unoriented percolation V der Hofstad–J´ arai 2004. Random Subgraphs Of Finite Graphs: The Phase Transition For The n-Cube. (Borgs–J. Chayes–v. der Hofstad-Slade–J. Spencer 2006) Lace expansion for the Ising model (high dimensions or finite range coupling). (A Sakai 2007)
Convergence of critical oriented percolation to super-Brownian motion above 4+1 dimensions. (van der Hofstad–Slade 2003). Incipient infinite cluster for spread-out oriented percolation above 4 + 1 dimensions (van der Hofstad–den Hollander–Slade 2002) Existence, properties of the incipient infinite cluster for high-dimensional unoriented percolation V der Hofstad–J´ arai 2004. Random Subgraphs Of Finite Graphs: The Phase Transition For The n-Cube. (Borgs–J. Chayes–v. der Hofstad-Slade–J. Spencer 2006) Lace expansion for the Ising model (high dimensions or finite range coupling). (A Sakai 2007) Lace expansion for the Ising model (high dimensions or finite range coupling). (A. Sakai 2007)
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