I Simposio de Postgrado 2023. Ingeniería, ciencias e innovación

MÓDULO_ 07 Matemáticas aplicadas y Modelación matemática 144 ABSTRACT Consider a model of N non-intersecting Brownian motions Modelamiento matemático Restricted maximum of non-intersecting Brownian bridges Yamit Yalanda 1* , Nicolás Zalduendo 2 1 Departamento de Ingeniería Matemática, Universidad de Chile. 2 Université de Lorraine, CNRS, Inria, IECL, UMR 7502, F-54000 Nancy, France. *Email: yamit@dim.uchile.cl Resumen Consider a model of non-intersecting Brownian motions for all started at zero at time 0 and ending at zero at time 1, and let us denote the maximal height achieved by the uppermost path within the interval by where is confined to the range [0,1]. This construction exhibits a fascinating behavior when becomes exceedingly large. It is known that, under a suitable rescaling, the distribution of converges, as , to a one- parameter family of distributions interpolating between the Tracy-Widom distributions for the Gaussian Orthogonal and Unitary Ensembles (corresponding, respectively, to and ). It is also known that, when the value of is fixed, has the distribution of the largest eigenvalue of a random matrix drawn from the Laguerre Orthogonal Ensemble. Here we show a version of these results for for fixed , showing that converges in distribution, as , to the rightmost charge in a generalized Laguerre Unitary Ensemble, which coincides with the top eigenvalue of a random matrix drawn from the Antisymmetric Gaussian Ensemble. N B 1 ( t ) ≤ B 2 ( t ) . . . ≤ B N ( t ) 0 ≤ t ≤ 1 [0, p ] M N ( p ) p N M N ( p ) N → ∞ p → 1 p → 0 N M N (1) M N ( p ) N M N ( p ) / p p → 0 for all started at zero at tim 0 and ending at zero at time 1 , and let us denote the maximal height achieved by the uppermost path within the interval [0, p ] by M N (p) where p is confined to the range [0,1] . This construction exhibits a fascinating behavior when N becomes exceedingly large. It is known that, under a suitable rescaling, the distribution of M N (p) converges, as N → ∞ , to a one-parameter family of distributions interpolating between the Tracy-Widom distributions for the Gaussian Orthogonal and Unitary Ensembles (corresponding, respectively, to p → 1 and p → 0 ). It is l o k own that, when the value of N is fixed, M N ( 1 ) has the distribution of the largest eigenvalue of a random matrix drawn from the Laguerre Orthogonal Ensemble. Here we show a version of these results for M N (p) for fixed N , showing that Modelamiento matemático Restricted maximum of non-intersecting Brownian bridges Yamit Yalanda 1* , Nicolás Zalduendo 2 1 Departamento de Ingeniería Matemática, Universi ad de Chile. 2 Université de Lorraine, CNRS, In ia, IECL, UMR 7502, F-54000 Nancy, Franc . *Email: yamit@dim.uchile.cl Resumen Consider a model of non-intersecting Brownian motions for all started at zero at time 0 and ending at zero at time 1, and let us denote the maximal height achieved by the uppermost path within the interval by where is confined to the range [0,1]. This construction exhibits a fascinating behavior when becomes exceedingly large. It is known that, under a suitable rescaling, the distribution of converges, as , to a one- parameter family of distributions interpolating between the Tracy-Widom distributions for the Gaussian Orthogonal and Unitary Ensembles (corresponding, respectively, to and ). It is also known that, when the value of is fixed, has the distribution of the largest eigenvalue of a random matrix drawn from the Laguerre Orthogonal Ensemble. Here we show a version of these results for for fixed , showing that converges in distribution, as , to the rightmost charge in a generalized Laguerre Unitary Ensemble, which coincides with the top eigenvalue of a random matrix drawn from the Antisymmetric Gaussian Ensemble. N B 1 ( t ) ≤ B 2 ( t ) . . . ≤ B N ( t ) 0 ≤ t ≤ 1 [0, p ] M N ( p ) p N M N ( p ) N → ∞ p → 1 p → 0 N M N (1) M N ( p ) N M N ( p ) / p p → 0 i distribution, as p → 0 , to the rightmost charge in a g neralized Lagu rre Unitary Ensemble, which coincides with the top igenvalue of a random matrix drawn from the Antisymmetric Gaussian Ensemble. Yamit Yalanda 1* , Nicolás Zalduendo 2 1 Departamento de Ingeniería Matemática, Universidad de Chile. 2 Université de Lorraine, CNRS, Inria, IECL, UMR 7502, F-54000 Nancy, France. *Email: yamit@dim.uchile.cl RESTRICTED MAXIMUM OF NON-INTERSECTING BROWNIAN BRIDGES