0 - General idea and limit object e Brownian excursion, S i.i.d. e ( x ) balanced signs indexed by the − local minima of e . Define a shuffled pseudo-order − + + on [ 0, 1 ] : x ⊳ S − e y if and only if x or ϕ ( x ) ⊕ ⊖ x y y x ϕ ( t ) = Leb ( { u ∈ [ 0, 1 ] , u ⊳ S e t } ) is the only (up to a.e. equality) Lebesgue-preserving function sending ≤ to ⊳ S e Then µ = ( id, ϕ ) ⋆ Leb is the Brownian separable permuton (M. 2017) x
I - Permuton convergence and patterns For σ ∈ S n and k ≤ n , perm k ( σ ) is a uniform subpermutation of length k in σ .
I - Permuton convergence and patterns For σ ∈ S n and k ≤ n , perm k ( σ ) is a uniform subpermutation of length k in σ . This notion is extended to permutons: perm k ( µ ) is the random permutation that is order-isomorphic to an i.i.d. pick according to µ .
I - Permuton convergence and patterns For σ ∈ S n and k ≤ n , perm k ( σ ) is a uniform subpermutation of length k in σ . This notion is extended to permutons: perm k ( µ ) is the random permutation that is order-isomorphic to an i.i.d. pick according to µ . Theorem (Hoppen et. al. ’2013, BBFGMP ’2017) The random permutons ( µ σ n ) converge in distribution to µ d iff for every k , perm k ( σ n ) − n → ∞ perm k ( µ ) . − − →
II - Patterns and the tree encoding A subpermutation of σ n can be read on a reduced tree of t n ⊖ ⊕ ⊖ ⊕
II - Patterns and the tree encoding A subpermutation of σ n can be read on a reduced tree of t n ⊖ ⊕ ⊖ ⊕
II - Patterns and the tree encoding A subpermutation of σ n can be read on a reduced tree of t n ⊖ ⊕ ⊖ ⊕
II - Patterns and the tree encoding A subpermutation of σ n can be read on a reduced tree of t n n ( σ n ) t n | I k pat I k n ⊕ ⊖ ⊖ ⊕ ⊖ ⊕
II - Patterns and the tree encoding A subpermutation of σ n can be read on a reduced tree of t n Consider a uniform k -reduced tree of a Schröder tree of size n . Here k = 3. I k n ( σ n ) t n | I k pat I k n n ⊕ ⊕ ⊖ ⊖ t n
II - Patterns and the tree encoding A subpermutation of σ n can be read on a reduced tree of t n Consider a uniform k -reduced tree of a Schröder tree of size n . Here k = 3. I k n ( σ n ) t n | I k pat I k n n ⊕ ⊕ ⊖ What does it look like as n → ∞ ? ⊖ t n
II - Patterns and the tree encoding A subpermutation of σ n can be read on a reduced tree of t n Consider a uniform k -reduced tree of a Schröder tree of size n . Here k = 3. I k pat I n ( σ n ) t n | I n n ⊕ ⊕ What does it look like as n → ∞ ? t n
III - Patterns in the Brownian permuton ϕ ( x ) − x + + − − − e ( x )
III - Patterns in the Brownian permuton Reduced trees of the Brownian ϕ ( x ) excursion are uniform binary trees (Aldous ’93, Le Gall ’93) − x + − b k e ( x )
III - Patterns in the Brownian permuton Reduced trees of the Brownian ϕ ( x ) excursion are uniform binary trees (Aldous ’93, Le Gall ’93) Hence perm k ( µ ) has the distribution of perm ( b k ) where b k is a uniform − x + signed binary tree − with k leaves. b k e ( x )
Summing up Fix a signed binary tree τ with k leaves. We need only show that # { Schröder trees of size n with k marked leaves inducing τ } # { Schröder trees of size n with k marked leaves } converges to 1 P ( b k = τ ) = . 2 k − 1 Cat k − 1
IV - Analytic combinatorics Let ( a n ) n be a nonnegative sequence and A ( z ) = ∑ n a n z n its generating function of radius ρ Transfer Theorem (Flajolet & Odlyzko) If • A is defined on a ∆ -domain at ρ > 0 (e.g. is algebraic) z → ρ g ( z ) + ( C + o ( 1 ))( ρ − z ) δ with g analytic, • A ( z ) = ∈ N , δ / Γ ( − δ ) + o ( 1 )) ρ − n n − 1 − δ C n → ∞ ( = then a n Proposition (Singular differentiation) Under the same hypotheses, A ′ ( z ) = z → ρ g ′ ( z ) + δ ( C + o ( 1 ))( ρ − z ) δ − 1
Analytic combinatorics for leaf-counted trees Recall: nice trees converge to the Brownian CRT.
Analytic combinatorics for leaf-counted trees Recall: nice trees converge to the Brownian CRT. Recursive trees counted by number of leaves. (Schröder: F ( t ) = ∑ k ≥ 2 t k ). T ( z ) = z + F ( T ( z ))
Analytic combinatorics for leaf-counted trees Recall: nice trees converge to the Brownian CRT. Recursive trees counted by number of leaves. (Schröder: F ( t ) = ∑ k ≥ 2 t k ). T ( z ) = z + F ( T ( z )) In this case, "very nice" if 0 < u < R F , F ′ ( u ) = 1. ∃ F(t) Then T is ∆ -analytic at ρ with T ( ρ ) = u and a square-root singularity (smooth implicit function schema). u t
Analytic combinatorics for leaf-counted trees Recall: nice trees converge to the Brownian CRT. Recursive trees counted by number of leaves. T ( z ) (Schröder: F ( t ) = ∑ k ≥ 2 t k ). T ( z ) = z + F ( T ( z )) In this case, "very nice" if 0 < u < R F , F ′ ( u ) = 1. ∃ F(t) Then T is ∆ -analytic at ρ with T ( ρ ) = u and a square-root singularity (smooth implicit function schema). u t z
Analytic combinatorics for leaf-counted trees Recall: nice trees converge to the Brownian CRT. Recursive trees counted by number of leaves. T ( z ) (Schröder: F ( t ) = ∑ k ≥ 2 t k ). T ( z ) = z + F ( T ( z )) u In this case, "very nice" if 0 < u < R F , F ′ ( u ) = 1. ∃ F(t) Then T is ∆ -analytic at ρ with T ( ρ ) = u and a square-root singularity (smooth implicit function schema). u t ρ z
Analytic combinatorics for leaf-counted trees Recall: nice trees converge to the Brownian CRT. Recursive trees counted by number of leaves. T ( z ) (Schröder: F ( t ) = ∑ k ≥ 2 t k ). T ( z ) = z + F ( T ( z )) u In this case, "very nice" if 0 < u < R F , F ′ ( u ) = 1. ∃ Then T is ∆ -analytic at ρ with T ( ρ ) = u and a square-root singularity (smooth implicit function schema). ρ z
Analytic combinatorics for leaf-counted trees Recall: nice trees converge to the Brownian CRT. Recursive trees counted by number of leaves. T ( z ) (Schröder: F ( t ) = ∑ k ≥ 2 t k ). T ( z ) = z + F ( T ( z )) u In this case, "very nice" if 0 < u < R F , F ′ ( u ) = 1. ∃ F(t) Then T is ∆ -analytic at ρ with T ( ρ ) = u and a square-root singularity (smooth implicit function schema). u t This is the case for Schröder ( F rational) ρ z
Uniform k -subtree in large unsigned trees T has square-root singularity at ρ and F analytic at T ( ρ ) . Then, the g . f of trees with k marked leaves that induce the k -tree τ is 1 z k T ′ ( z ) T ′ ( z ) deg ( v ) deg ( v ) ! F ( deg ( v )) ( T ( z )) ∏ v internal node of τ τ
Uniform k -subtree in large unsigned trees T has square-root singularity at ρ and F analytic at T ( ρ ) . Then, the g . f of trees with k marked leaves that induce the k -tree τ is 1 z k T ′ ( z ) T ′ ( z ) deg ( v ) deg ( v ) ! F ( deg ( v )) ( T ( z )) ∏ v internal node of τ τ
Uniform k -subtree in large unsigned trees T has square-root singularity at ρ and F analytic at T ( ρ ) . Then, the g . f of trees with k marked leaves that induce the k -tree τ is 1 z k T ′ ( z ) T ′ ( z ) deg ( v ) deg ( v ) ! F ( deg ( v )) ( T ( z )) ∏ v internal node of τ τ
Uniform k -subtree in large unsigned trees T has square-root singularity at ρ and F analytic at T ( ρ ) . Then, the g . f of trees with k marked leaves that induce the k -tree τ is 1 z k T ′ ( z ) T ′ ( z ) deg ( v ) deg ( v ) ! F ( deg ( v )) ( T ( z )) ∏ v internal node of τ τ zT ′ zT ′ zT ′ F ( 3 ) 3! ( T ) T ′
Uniform k -subtree in large unsigned trees T has square-root singularity at ρ and F analytic at T ( ρ ) . Then, the g . f of trees with k marked leaves that induce the k -tree τ is 1 z k T ′ ( z ) T ′ ( z ) deg ( v ) deg ( v ) ! F ( deg ( v )) ( T ( z )) ∏ v internal node of τ ∼ ρ C τ ( ρ − z ) − # { nodes in τ } /2 . Dominates when τ binary. τ (Then C τ doesn’t depend on zT ′ zT ′ zT ′ τ ). Transfer: t n | I k n converges in distribution to a uniform F ( 3 ) binary tree. 3! ( T ) T ′
Uniform k -subtree in large signed trees Counting signed trees that induce a given signed tree τ : adding parity constraints on the height of the marked leaf in the marked trees.
Uniform k -subtree in large signed trees Counting signed trees that induce a given signed tree τ : adding parity constraints on the height of the marked leaf in the marked trees. Replace instances of T ′ by T ′ 0 (even height) or T ′ 1 (odd 1 = T ′ and T ′ height). T ′ 0 + T ′ 1 = F ′ ( T ) T ′ 0 , so T ′ 0 ∼ T ′ 1 ∼ 1 2 T ′ .
Uniform k -subtree in large signed trees Counting signed trees that induce a given signed tree τ : adding parity constraints on the height of the marked leaf in the marked trees. Replace instances of T ′ by T ′ 0 (even height) or T ′ 1 (odd 1 = T ′ and T ′ height). T ′ 0 + T ′ 1 = F ′ ( T ) T ′ 0 , so T ′ 0 ∼ T ′ 1 ∼ 1 2 T ′ . g.f. of Trees with k marked leaves that induce the signed k -tree τ : 1 b T ′ a T ′ k z k ( T ′ 0 + T ′ 1 ) T ′ deg ( v ) ! F ( deg ( v )) ( T ( z )) ∏ 0 1 v internal node of τ where a (resp. b ) is the number of edges of τ incident to two nodes of the same (resp. different) signs
Uniform k -subtree in large signed trees Counting signed trees that induce a given signed tree τ : adding parity constraints on the height of the marked leaf in the marked trees. Replace instances of T ′ by T ′ 0 (even height) or T ′ 1 (odd 1 = T ′ and T ′ height). T ′ 0 + T ′ 1 = F ′ ( T ) T ′ 0 , so T ′ 0 ∼ T ′ 1 ∼ 1 2 T ′ . g.f. of Trees with k marked leaves that induce the signed k -tree τ : 1 b T ′ a T ′ k z k ( T ′ 0 + T ′ 1 ) T ′ deg ( v ) ! F ( deg ( v )) ( T ( z )) ∏ 0 1 v internal node of τ where a (resp. b ) is the number of edges of τ incident to two nodes of the same (resp. different) signs Hence all signed binary trees have the same asymptotic probability, what whe needed for permuton convergence.
Part 2 - statement
Substitution decomposition For σ ∈ S k , ρ 1 , . . . , ρ k ∈ S , define σ [ ρ 1 , . . . , ρ k ] by replacing the i -th dot in σ by π i . Example : 132 [ 21, 312, 2413 ] = 219784635.
Substitution decomposition For σ ∈ S k , ρ 1 , . . . , ρ k ∈ S , define σ [ ρ 1 , . . . , ρ k ] by replacing the i -th dot in σ by π i . Example : 132 [ 21, 312, 2413 ] = 219784635.
Substitution decomposition For σ ∈ S k , ρ 1 , . . . , ρ k ∈ S , define σ [ ρ 1 , . . . , ρ k ] by replacing the i -th dot in σ by π i . Example : 132 [ 21, 312, 2413 ] = 219784635.
Substitution decomposition For σ ∈ S k , ρ 1 , . . . , ρ k ∈ S , define σ [ ρ 1 , . . . , ρ k ] by replacing the i -th dot in σ by π i . Example : 132 [ 21, 312, 2413 ] = 219784635.
Substitution decomposition For σ ∈ S k , ρ 1 , . . . , ρ k ∈ S , define σ [ ρ 1 , . . . , ρ k ] by replacing the i -th dot in σ by π i . Example : 132 [ 21, 312, 2413 ] = 219784635. ⊕ and ⊖ are just substitutions into increasing and decreasing permutations
Substitution decomposition For σ ∈ S k , ρ 1 , . . . , ρ k ∈ S , define σ [ ρ 1 , . . . , ρ k ] by replacing the i -th dot in σ by π i . Example : 132 [ 21, 312, 2413 ] = 219784635. ⊕ and ⊖ are just substitutions into increasing and decreasing permutations Given σ , either :
Substitution decomposition For σ ∈ S k , ρ 1 , . . . , ρ k ∈ S , define σ [ ρ 1 , . . . , ρ k ] by replacing the i -th dot in σ by π i . Example : 132 [ 21, 312, 2413 ] = 219784635. ⊕ and ⊖ are just substitutions into increasing and decreasing permutations Given σ , either : • We can find a proper interval mapped to an interval, and then σ can be written as a substitution of smaller permutations
Substitution decomposition For σ ∈ S k , ρ 1 , . . . , ρ k ∈ S , define σ [ ρ 1 , . . . , ρ k ] by replacing the i -th dot in σ by π i . Example : 132 [ 21, 312, 2413 ] = 219784635. ⊕ and ⊖ are just substitutions into increasing and decreasing permutations Given σ , either : • We can find a proper interval mapped to an interval, and then σ can be written as a substitution of smaller permutations • Or σ can’t be decomposed by a nontrivial substitution : σ is a simple permutation . Ex : 1, 12, 21, 2413, 3142, 31524, ... ∼ n ! e 2 .
Substitution decomposition ( 8, 10, 9, 2, 11, 1, 4, 7, 3, 6, 5 )
Substitution decomposition ( 8, 10, 9, 2, 11, 1, 4, 7, 3, 6, 5 )
Substitution decomposition ( 8, 10, 9, 2, 11, 1, 4, 7, 3, 6, 5 )
Substitution decomposition ( 8, 10, 9, 2, 11, 1, 4, 7, 3, 6, 5 )
Substitution decomposition ( 8, 10, 9, 2, 11, 1, 4, 7, 3, 6, 5 )
Substitution decomposition ( 8, 10, 9, 2, 11, 1, 4, 7, 3, 6, 5 )
Substitution decomposition ( 8, 10, 9, 2, 11, 1, 4, 7, 3, 6, 5 )
Substitution decomposition ( 8, 10, 9, 2, 11, 1, 4, 7, 3, 6, 5 )
Substitution decomposition ( 8, 10, 9, 2, 11, 1, 4, 7, 3, 6, 5 )
Substitution decomposition ( 8, 10, 9, 2, 11, 1, 4, 7, 3, 6, 5 )
Substitution decomposition ( 8, 10, 9, 2, 11, 1, 4, 7, 3, 6, 5 ) ⊖ ⊖ 2413 ⊕ 42513
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