Erratum to “Non-intersecting Brownian walkers and Yang–Mills theory on the sphere” [Nucl. Phys. B 844 (3) (2011) 500

Abstract

The paper [1] deals with the study of N non-intersecting Brownian motions on a line segment [0,L] with three different types of boundary conditions: (I) periodic, (II) absorbing and (III) reflecting boundary conditions. We showed in particular that the normalized reunion probabilities of these Brownian motions in the three models can be mapped to the partition function of two-dimensional continuum Yang–Mills theory on a sphere respectively with gauge groups U(N), Sp(2N) and SO(2N). Consequently, we showed that in each of these Brownian motion models, as one varies the system size L, a third order phase transition occurs at a critical value L= Lc(N)∼ √ N in the large N limit. Close to the critical point, the reunion probability, properly centered and scaled, can be expressed in terms of Tracy–Widom distributions describing the probability distribution of the largest eigenvalue of random matrices. While our statements for case (I) and case (II), the later being the most interesting one, are perfectly correct, the results for the rescaled reunion probability ẼN (L) in the limit of large N for the case (III), which we only discussed briefly in Ref. [1], is actually erroneous: this is the reason for this erratum. We wrote indeed in Ref. [1] that ẼN(L) converges, when N →∞, to the Tracy–Widom distribution corresponding to GOE random matrices. We correct this wrong statement and find instead that ẼN(L) converges, when N →∞, to the ratio of F2(t)/F1(t) where F2(t) and F1(t) are respectively the Tracy–Widom distributions corresponding respectively to GUE and GOE random matrices, in terms of the variable t = 211/6|L−√2N |.