Expected maximum of bridge random walks & Lévy flights

Abstract

Abstract We consider one-dimensional discrete-time random walks (RWs) with arbitrary symmetric and continuous jump distributions f ( η ), including the case of Lévy flights. We study the expected maximum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="double-struck">E</mml:mi> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> </mml:math> of bridge RWs, i.e. RWs starting and ending at the origin after n steps. We obtain an exact analytical expression for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="double-struck">E</mml:mi> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> </mml:math> valid for any n and jump distribution f ( η ), which we then analyze in the large n limit up to second leading order term. For jump distributions whose Fourier transform behaves, for small k , as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> <mml:mo stretchy="false">^</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mo>∼</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mo stretchy="false">|</mml:mo> <mml:mi>a</mml:mi> <mml:mi>k</mml:mi> <mml:msup> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>μ</mml:mi> </mml:mrow> </mml:msup> </mml:math> with a Lévy index 0 &lt; μ ⩽ 2 and an arbitrary length scale a &gt; 0, we find that, at leading order for large n , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="double-struck">E</mml:mi> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:mo>∼</mml:mo> <mml:mi>a</mml:mi> <mml:msub> <mml:mrow> <mml:mi>h</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>μ</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:msup> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>μ</mml:mi> </mml:mrow> </mml:msup> </mml:math> . We obtain an explicit expression for the amplitude h 1 ( μ ) and find that it carries the signature of the bridge condition, being different from its counterpart for the free RW. For μ = 2, we find that the second leading order term is a constant, which, quite remarkably, is the same as its counterpart for the free RW. For generic 0 &lt; μ &lt; 2, this second leading order term is a growing function of n , which depends non-trivially on further details of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> <mml:mo stretchy="false">^</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , beyond the Lévy index μ . Finally, we apply our results to compute the mean perimeter of the convex hull of the 2 d Rouse polymer chain and of the 2 d run-and-tumble particle, as well as to the computation of the survival probability in a bridge version of the well-known ‘lamb–lion’ capture problem.

Topics & Concepts