Asymptotics for the number of walks in a Weyl chamber of type B. Issue 2 (8th October 2012)

• Record Type: Journal Article
• Authors: Feierl, Thomas
• Is Part Of: Random structures & algorithms. Volume 45:Issue 2(2014)
• Page Start: 261
• Page End: 305
• Publication Date: 2012-10-08
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20467
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>ABSTRACT</title> <p>We consider lattice walks in <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghv4p042p" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20467:rsa20467-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mo>ℝ</mml:mo><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:math></alternatives></inline-formula> confined to the region <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghv4p0415" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20467:rsa20467-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo><</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>…</mml:mo><mml:mo><</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula> with fixed (but arbitrary) starting and end points. These walks are assumed to be such that their number can be counted using a reflection principle argument. The main results are asymptotic formulas for the total number of walks of length <italic>n</italic> with either a fixed or a free end point for a general class of walks as <italic>n</italic> tends to infinity. As applications, we find the asymptotics for the number of <italic>k</italic>‐non‐crossing tangled diagrams as well as asymptotics for two <italic>k</italic>‐vicious walkers models subject to a wall restriction. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 261–305, 2014</p> </abstract>