Walks with large steps in the quadrant Mireille Bousquet-Mélou, CNRS, Université de Bordeaux based on work with Alin Bostan, INRIA Saclay, Paris Steve Melczer, University of Waterloo and École normale supérieure de Lyon
Outline I. Motivation II. A general approach... that solves some cases III. What can go wrong? IV. Some cases that work
Counting quadrant walks Let S be a finite subset of Z 2 (set of steps) and p 0 ∈ N 2 (starting point). Example. S = { 10 , ¯ 10 , 1 ¯ 1 , ¯ 11 } , p 0 = ( 0 , 0 )
Counting quadrant walks Let S be a finite subset of Z 2 (set of steps) and p 0 ∈ N 2 (starting point). A path (walk) of length n starting at p 0 is a sequence ( p 0 , p 1 , . . . , p n ) such that p i + 1 − p i ∈ S for all i . Example. S = { 10 , ¯ 10 , 1 ¯ 1 , ¯ 11 } , p 0 = ( 0 , 0 )
Counting quadrant walks Let S be a finite subset of Z 2 (set of steps) and p 0 ∈ N 2 (starting point). A path (walk) of length n starting at p 0 is a sequence ( p 0 , p 1 , . . . , p n ) such that p i + 1 − p i ∈ S for all i . What is the number q ( n ) of n -step walks starting at p 0 and contained in N 2 ? For ( i , j ) ∈ N 2 , what is the number q ( i , j ; n ) of such walks that end at ( i , j ) ? Example. S = { 10 , ¯ 10 , 1 ¯ 1 , ¯ 11 } , p 0 = ( 0 , 0 ) ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ( i , j ) = ( 5 , 1 ) ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� �����������
Counting quadrant walks: higher dimension Let S be a finite subset of Z d (set of steps) and p 0 ∈ N d (starting point). A path (walk) of length n starting at p 0 is a sequence ( p 0 , p 1 , . . . , p n ) such that p i + 1 − p i ∈ S for all i . What is the number q ( n ) of n -step walks starting at p 0 and contained in N d ? For i = ( i 1 , . . . , i d ) ∈ N d , what is the number q ( i ; n ) of such walks that end at i ? Example. S = { 10 , ¯ 10 , 1 ¯ 1 , ¯ 11 } , p 0 = ( 0 , 0 ) ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ( i , j ) = ( 5 , 1 ) ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� �����������
Counting quadrant walks: higher dimension Let S be a finite subset of Z d (set of steps) and p 0 ∈ N d (starting point). A path (walk) of length n starting at p 0 is a sequence ( p 0 , p 1 , . . . , p n ) such that p i + 1 − p i ∈ S for all i . What is the number q ( n ) of n -step walks starting at p 0 and contained in N d ? For i = ( i 1 , . . . , i d ) ∈ N d , what is the number q ( i ; n ) of such walks that end at i ? The associated generating function: 1 · · · x i d � � q ( i 1 , . . . , i d ; n ) x i 1 d t n Q ( x 1 , . . . , x d ; t ) = n ≥ 0 ( i 1 ,..., i d ) ∈ N d What is the nature of this series?
A hierarchy of formal power series • The formal power series A ( t ) is rational if it can be written A ( t ) = P ( t ) / Q ( t ) where P ( t ) and Q ( t ) are polynomials in t .
A hierarchy of formal power series • The formal power series A ( t ) is rational if it can be written A ( t ) = P ( t ) / Q ( t ) where P ( t ) and Q ( t ) are polynomials in t . • The formal power series A ( t ) is algebraic (over Q ( t ) ) if it satisfies a (non-trivial) polynomial equation: P ( t , A ( t )) = 0 .
A hierarchy of formal power series • The formal power series A ( t ) is rational if it can be written A ( t ) = P ( t ) / Q ( t ) where P ( t ) and Q ( t ) are polynomials in t . • The formal power series A ( t ) is algebraic (over Q ( t ) ) if it satisfies a (non-trivial) polynomial equation: P ( t , A ( t )) = 0 . • The formal power series A ( t ) is D-finite (holonomic) if it satisfies a (non-trivial) linear differential equation with polynomial coefficients: P k ( t ) A ( k ) ( t ) + · · · + P 0 ( t ) A ( t ) = 0 .
A hierarchy of formal power series • The formal power series A ( t ) is rational if it can be written A ( t ) = P ( t ) / Q ( t ) where P ( t ) and Q ( t ) are polynomials in t . • The formal power series A ( t ) is algebraic (over Q ( t ) ) if it satisfies a (non-trivial) polynomial equation: P ( t , A ( t )) = 0 . • The formal power series A ( t ) is D-finite (holonomic) if it satisfies a (non-trivial) linear differential equation with polynomial coefficients: P k ( t ) A ( k ) ( t ) + · · · + P 0 ( t ) A ( t ) = 0 . ◦ Nice closure properties + asymptotics of the coefficients
A hierarchy of formal power series • The formal power series A ( t ) is rational if it can be written A ( t ) = P ( t ) / Q ( t ) where P ( t ) and Q ( t ) are polynomials in t . • The formal power series A ( t ) is algebraic (over Q ( t ) ) if it satisfies a (non-trivial) polynomial equation: P ( t , A ( t )) = 0 . • The formal power series A ( t ) is D-finite (holonomic) if it satisfies a (non-trivial) linear differential equation with polynomial coefficients: P k ( t ) A ( k ) ( t ) + · · · + P 0 ( t ) A ( t ) = 0 . ◦ Nice closure properties + asymptotics of the coefficients ◦ Extension to several variables (D-finite: one DE per variable)
Classification of quadrant walks with small steps Theorem Assume S ⊂ { ¯ 1 , 0 , 1 } 2 . The series Q ( x , y ; t ) is D-finite iff a certain group G associated with S is finite. It is algebraic iff, in addition, the “orbit sum” is zero. [mbm-Mishna 10], [Bostan-Kauers 10] D-finite [Kurkova-Raschel 12] non-singular non-D-finite [Mishna-Rechnitzer 07], [Melczer-Mishna 13] singular non-D-finite quadrant models with small steps: 79 | G | < ∞ : 23 | G | = ∞ : 56 OS � = 0: 19 OS = 0: 4 Not D-finite D-finite algebraic
Classification of quadrant walks with small steps quadrant models with small steps: 79 | G | < ∞ : 23 | G | = ∞ : 56 OS � = 0: 19 OS = 0: 4 Not D-finite D-finite algebraic Random walks in probability Formal power series algebra D-finite series effective closure properties Computer algebra arithmetic properties asymptotics Complex analysis G-functions
Quadrant walks with large steps A mathematical challenge: the small step condition seems crucial in all approaches (apart from computer algebra) Is the nice classification of walks with small steps robust? Large steps occur in “real life”: bipolar orientations of regular maps [Kenyon, Miller, Sheffield, Wilson, 15(a)] N S
From quadrant walks to bipolar regular maps [KMSW 15(a)] Fix p ≥ 1, and take a quadrant walk with two kinds of steps: SE steps ( 1 , − 1 ) NW steps ( − i , j ) with i , j ≥ 0 and i + j = p The construction starts from a quadrant excursion and a bipolar map reduced to an edge, and yields a bipolar map with faces of degree p + 2. Ex: p = 2 N S
From quadrant walks to bipolar regular maps [KMSW 15(a)] The construction starts from a quadrant excursion and a bipolar map reduced to an edge, and yields a bipolar map with faces of degree p + 2. every SE step ( 1 , − 1 ) creates an edge. every NW step ( − i , j ) creates a face of degree i + j + 2 and an edge. ( 1 , − 1 ) or ( − i , j ) i + 1 j + 1
From quadrant walks to bipolar regular maps [KMSW 15(a)] Proposition [Kenyon et al. 15(a)] This construction is a bijection from quadrant excursions to bipolar maps with faces of degree p + 2. N S • steps ⇔ edges in the orientation (minus 1)
II. A general approach for quadrant walks... which solves some cases. quadrant models with small steps: 79 | G | < ∞ : 23 | G | = ∞ : 56 OS � = 0: 19 OS = 0: 4 Not D-finite D-finite algebraic
A four step approach 1. Write a functional equation for the tri-variate series Q ( x , y ; t ) . It will involve bi-variate series Q ( x , 0 ; t ) , Q ( 0 , y ; t ) , . . . (called sections) 2. Compute the “orbit” of ( x , y ) 3. Find a functional equation free from sections 4. Extract from it Q ( x , y ; t )
Step 1: Write a functional equation Example: S = { 01 , ¯ 10 , 1 ¯ 1 } (bipolar triangulations) Q ( x , y ; t ) ≡ Q ( x , y ) = 1 + t ( y + ¯ xQ ( 0 , y ) − tx ¯ yQ ( x , 0 ) x + x ¯ y ) Q ( x , y ) − t ¯ with ¯ x = 1 / x and ¯ y = 1 / y .
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