TY - JOUR

T1 - Winding number correlation for a Brownian loop in a plane

AU - Hannay, J. H.

PY - 2019/1/18

Y1 - 2019/1/18

N2 - A Brownian loop is a random walk circuit of infinitely many, suitably infinitesimal, steps. In a plane such a loop may or may not enclose a marked point, the origin, say. If it does so it may wind arbitrarily many times, positive or negative, around that point. Indeed, from the (long known) probability distribution, the mean square winding number is infinite, so all statistical moments—averages of powers of the winding number—are infinity (even powers) or zero (odd powers, by symmetry). If an additional marked point is introduced at some distance from the origin, there are now two winding numbers, which are correlated. That correlation, the average of the product of the two winding numbers, is finite, and is calculated here. The result takes the form of a single well-convergent integral that depends on a single parameter—the suitably scaled separation of the marked points. The integrals of the correlation weighted by powers of the separation are simple factorial expressions. Explicit limits of the correlation for small and large separation of the marked points are found.

AB - A Brownian loop is a random walk circuit of infinitely many, suitably infinitesimal, steps. In a plane such a loop may or may not enclose a marked point, the origin, say. If it does so it may wind arbitrarily many times, positive or negative, around that point. Indeed, from the (long known) probability distribution, the mean square winding number is infinite, so all statistical moments—averages of powers of the winding number—are infinity (even powers) or zero (odd powers, by symmetry). If an additional marked point is introduced at some distance from the origin, there are now two winding numbers, which are correlated. That correlation, the average of the product of the two winding numbers, is finite, and is calculated here. The result takes the form of a single well-convergent integral that depends on a single parameter—the suitably scaled separation of the marked points. The integrals of the correlation weighted by powers of the separation are simple factorial expressions. Explicit limits of the correlation for small and large separation of the marked points are found.

KW - Brownian loop

KW - Correlation

KW - Winding number

UR - http://www.scopus.com/inward/record.url?scp=85061938562&partnerID=8YFLogxK

U2 - 10.1088/1751-8121/aaea03

DO - 10.1088/1751-8121/aaea03

M3 - Article (Academic Journal)

AN - SCOPUS:85061938562

VL - 52

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 6

M1 - 065001

ER -