Brownian Motion Area with Generatingfunctionology Uwe Schwerdtfeger - PowerPoint PPT Presentation

Introduction and Results Proofs Applications Brownian Motion Area with Generatingfunctionology Uwe Schwerdtfeger RMIT University/University of Melbourne Alexander von Humboldt Foundation Monash University 3 August, 2011 U. Schwerdtfeger Brownian Motion Area

Introduction and Results Proofs Applications Some continuous time processes... A Brownian Motion of duration 1 is a stochastic process B ( t ) , t ∈ [0 , 1] such that ◮ t �→ B ( t ) is a.s. continuous, B (0) = 0 , ◮ for s < t , B ( t ) − B ( s ) ∼ N (0 , t − s ) and ◮ increments are independent. A Brownian Meander M ( t ) , t ∈ [0 , 1] is a BM B ( t ) conditioned on B ( s ) ≥ 0 , s ∈ ]0 , 1] . A Brownian Excursion E ( t ) , t ∈ [0 , 1] is M ( t ) conditioned on M (1) = 0 (quick and dirty def.). U. Schwerdtfeger Brownian Motion Area

Introduction and Results Proofs Applications ... and their discrete counterparts The Bernoulli Random Walk B n ( k ) on Z , k ∈ { 0 , 1 , . . . , n } , with ◮ B n (0) = 0 , ◮ B n ( k + 1) − B n ( k ) ∈ {− 1 , 1 } , each with prob. 1 / 2 . The Bernoulli Meander M n ( k ) , k ∈ { 0 , . . . , n } on Z ≥ 0 is B n ( k ) conditioned to stay non-negative. The Bernoulli Excursion E 2 n ( k ) , k ∈ { 0 , . . . , 2 n } on Z ≥ 0 is M 2 n ( k ) conditioned on M 2 n (2 n ) = 0 . U. Schwerdtfeger Brownian Motion Area

Introduction and Results Proofs Applications Scaling limits For n − → ∞ we have the weak limits � � 1 √ n B n ( ⌊ nt ⌋ ) , t ∈ [0 , 1] − → {B ( t ) , t ∈ [0 , 1] } , ◮ � � 1 √ n M n ( ⌊ nt ⌋ ) , t ∈ [0 , 1] − → {M ( t ) , t ∈ [0 , 1] } , ◮ � � 1 2 n E 2 n ( ⌊ 2 nt ⌋ ) , t ∈ [0 , 1] − → {E ( t ) , t ∈ [0 , 1] } . ◮ √ U. Schwerdtfeger Brownian Motion Area

Introduction and Results Proofs Applications Drmota (2003): Weak limits imply moment convergence for certain functionals. E.g. for area (i.e. integrals) ��� 1 � r � 1 → E [ EA r ] , √ E 2 n ( ⌊ 2 nt ⌋ ) dt − E 2 n 0 ��� 1 � r � 1 → E [ MA r ] , √ n M n ( ⌊ nt ⌋ ) dt − E 0 for n − → ∞ , where � 1 � 1 EA := E ( t ) dt , MA := M ( t ) dt . 0 0 So studying functionals on E or M amounts to studying the discrete models! U. Schwerdtfeger Brownian Motion Area

Introduction and Results Proofs Applications Particularly EA appears in a number of discrete contexts, e.g. ◮ Construction costs of hash tables, ◮ cost of breadth first search traversal of a random tree, ◮ path lengths in random trees, ◮ area of polyominoes, ◮ enumeration of connected graphs. Many of the discrete results rely on recursions for the moments of EA and MA found by Tak´ acs (1991,1995) studying E 2 n and M n . U. Schwerdtfeger Brownian Motion Area

Introduction and Results Proofs Applications Results We choose a different combinatorial approach and obtain ◮ new formulae for E ( EA r ) and E ( MA r ) , ◮ the joint distribution of ( MA , M (1)) in terms of the joint moments E ( MA r M (1) s ) , ◮ the joint distribution of (signed) areas and endpoint of B , and as an application of these ◮ area of discrete meanders with arbitrary finite step sets, ◮ area distribution of column convex polyominoes. U. Schwerdtfeger Brownian Motion Area

Introduction and Results Proofs Applications In the discrete world, we can write the joint distribution of the random variables n � M n ( k ) and H n = M n ( n ) A n = k =0 as p n , k , l P ( A n = k , H n = l ) = , � r , s p n , r , s where p n , k , l is the number of meanders of length n , area k and final height l . U. Schwerdtfeger Brownian Motion Area

Introduction and Results Proofs Applications The generating function of the class of meanders is the formal power series   � � p n , k , l q k u l  z n , M ( z , q , u ) = n k , l The above probabilities can be rewritten as p n , k , l P ( A n = k , H n = l ) = � r , s p n , r , s � z n q k u l � M ( z , q , u ) = . [ z n ] M ( z , 1 , 1) U. Schwerdtfeger Brownian Motion Area

Introduction and Results Proofs Applications   � � p n , k , l q k u l  z n , M ( z , q , u ) = n k , l and � z n q k u l � M ( z , q , u ) P ( A n = k , H n = l ) = . [ z n ] M ( z , 1 , 1) With this representation the moments take a particularly nice form: � E ( A r n H s k r l s P ( A n = k , H n = l ) n ) = k , l � r � � s M ( z , 1 , 1) � [ z n ] q ∂ u ∂ ∂ q ∂ u = . [ z n ] M ( z , 1 , 1) So: large n behaviour of the moments by coefficient asymptotics of the above series. U. Schwerdtfeger Brownian Motion Area

Introduction and Results Proofs Applications Singularity analysis (Flajolet, Odlyzko 1990) Transfer Theorem: Let F ( z ) = � f n z n be analytic in an indented disk and F ( z ) ∼ (1 − µ z ) − α ( z − → 1 /µ ) . Then 1 f n ∼ [ z n ] (1 − µ z ) − α ∼ Γ( α ) × n α − 1 × µ n ( n − → ∞ ) . For example, it turns out, that � ∂ � r � ∂ � s b r , s M ( z , 1 , 1) ∼ ( z − → 1 / 2) , ∂ q ∂ u (1 − 2 z ) 3 r / 2+ s / 2+1 / 2 U. Schwerdtfeger Brownian Motion Area

Introduction and Results Proofs Applications Functional equation for M ( z , q , u ) . The recursive description of the set of meanders { meanders of length n } ≃ { meanders of length n − 1 } × {ր , ց} \ { excursions of length n − 1 } × {ց} translates into � zuq + z � − E ( z , q ) z M ( z , q , u ) = 1 + M ( z , q , uq ) uq , uq E ( z , q ) is the generating function of excursions. U. Schwerdtfeger Brownian Motion Area

Introduction and Results Proofs Applications Solution to the equation for q = 1 by the kernel method: − z ( u − u 1 ( z ))( u − v 1 ( z )) M ( z , 1 , u ) = u − zE ( z , 1) . √ √ 1 − 4 z 2 1 − 4 z 2 where u 1 ( z ) = 1 − and v 1 ( z ) = 1+ . 2 z 2 z Substitution of u = u 1 ( z ) yields √ 1 − 4 z 2 E ( z , 1) = u 1 ( z ) = 1 − , 2 z 2 z and finally 1 M ( z , 1 , u ) = − z ( u − v 1 ( z )) . U. Schwerdtfeger Brownian Motion Area

Introduction and Results Proofs Applications � r � ∂ � s M ( z , 1 , u ) can in principle be � ∂ The partial derivatives ∂ q ∂ u obtained inductively by taking derivatives of the functional equation (and setting q = 1). ◮ Each derivative w.r.t. u produces one new unknown function � r � ∂ � s +1 M ( z , 1 , u ) . � ∂ ∂ q ∂ u ◮ Each derivative w.r.t. q produces two new unknowns, � r +1 � r +1 � ∂ � s M ( z , 1 , u ) and hence � � ∂ ∂ E ( z , 1) and ∂ q ∂ q ∂ u requires another application of the kernel method. U. Schwerdtfeger Brownian Motion Area

Introduction and Results Proofs Applications � r � ∂ � s M ( z , 1 , u ) and for � ∂ The exact expressions for ∂ q ∂ u � r � � s M ( z , 1 , 1) � q ∂ u ∂ [ z n ] ∂ q ∂ u E ( A r n H s n ) = . [ z n ] M ( z , 1 , 1) are getting intractable. But we can keep track of the singular behaviour of � r � ∂ � r � ∂ � s M ( z , 1 , 1) and � s M ( z , 1 , u 1 ( z )) and via � � ∂ ∂ ∂ q ∂ u ∂ q ∂ u singularity analysis large n asymptotics for the moments. U. Schwerdtfeger Brownian Motion Area

Introduction and Results Proofs Applications One proceeds in two steps: First show by induction � ∂ � r � ∂ � s a r , s M ( z , 1 , u 1 ( z )) ∼ ( z − → 1 / 2) , (1 − 2 z ) 3 r / 2+ s / 2+1 / 2 ∂ q ∂ u where a r , s = a r , s − 1 + ( s + 2) a r − 1 , s +2 , and then by induction � ∂ � r � ∂ � s b r , s M ( z , 1 , 1) ∼ ( z − → 1 / 2) , (1 − 2 z ) 3 r / 2+ s / 2+1 / 2 ∂ q ∂ u where b r , s = b r , s − 2 + ( s + 1) b r − 1 , s +1 , ( s ≥ 1) , b r , 0 = b r − 1 , 1 + a r − 1 , 1 . U. Schwerdtfeger Brownian Motion Area

Introduction and Results Proofs Applications Application of the transfer theorem finally yields: Γ(1 / 2) n ) ∼ b r , s Γ((3 r + s ) / 2) n (3 r + s ) / 2 , E ( A r n H s b 0 , 0 and hence (after rescaling n − 3 / 2 A n and n − 1 / 2 H n ) ◮ b r , s is essentially E ( MA r M (1) s ) , ◮ similarly a r − 1 , 1 is essentially E ( EA r ) . U. Schwerdtfeger Brownian Motion Area

Introduction and Results Proofs Applications Discrete meanders and excursions with arbitrary finite step sets: No result on convergence to M resp. E ! But: ◮ Generating function satisfies a similar functional equation. ◮ Area moments for meanders and excursions can be computed in the same fashion, ◮ and are expressed in terms of the very same b r , s resp. a r − 1 , 1 ! Result depends on the sign of the drift = mean of the step set. U. Schwerdtfeger Brownian Motion Area

Introduction and Results Proofs Applications Column convex polyominoes: Area distribution on polyominoes with fixed perimeter n . ◮ Similar functional equation as above. ◮ Similar arguments yield an EA limit law as n − → ∞ . U. Schwerdtfeger Brownian Motion Area

Introduction and Results Proofs Applications Acknowledgement Thank you for your attention! And thanks to ◮ Alexander von Humboldt Foundation ◮ RMIT University ◮ The ARC Centre of Excellence Mathematics and Statistics of Complex Systems (MASCOS) for financial support. U. Schwerdtfeger Brownian Motion Area