We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 25 June 2025
Selecting N random points in a unit square corresponds to selecting a random permutation. Placing symmetry restrictions on the points, we obtain special kinds of permutations: involutions, signed permutations and signed involutions. We are interested in the statistics of the length (in numbers of points) of the longest up/right path in each symmetry type as the number of points increases to infinity. The limiting distribution functions are expressed in terms of a Painleve II equation. In addition to the Tracy-Widom distributions of random matrix theory, we also obtain two new classes of distribution functions interpolating between the GOE and GSE, and between the GUE and GOE2 Tracy-Widom distribution functions. Applications to random vicious walks and site percolation are also discussed
1. Introduction
Suppose that we are selecting n points, p1 , p 2 , … , pn , at random in a rectangle, say R = [0,1] x [0,1] (see Figure 1). We denote by π the configuration of n random points. With probability 1, no two points have same x -coordinates nor -coordinates. An up/right path of π is a collection of points pi1, pi2, … ,pik such that x(pil) < x(pi2) < • • • < x(pik) and y(pi1)< y(pi2) < • • • < y (pik). The length of such a path is defined by the number of the points in the path. Now we denote by ln(π) the length of the longest up/right path of a random points configuration π.
To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.
