  As an image of the many-to-one map of loop-erasing operation $\LE$ of random
walks, a self-avoiding walk (SAW) is obtained. The loop-erased random walk
(LERW) model is the statistical ensemble of SAWs such that the weight of each
SAW $\zeta$ is given by the total weight of all random walks $\pi$ which are
inverse images of $\zeta$, $\{\pi: \LE(\pi)=\zeta \}$. We regard the Brownian
paths as the continuum limits of random walks and consider the statistical
ensemble of loop-erased Brownian paths (LEBPs) as the continuum limits of the
LERW model. Following the theory of Fomin on nonintersecting LERWs, we
introduce a nonintersecting system of $N$-tuples of LEBPs in a domain $D$ in
the complex plane, where the total weight of nonintersecting LEBPs is given by
Fomin's determinant of an $N \times N$ matrix whose entries are boundary
Poisson kernels in $D$. We set a sequence of chambers in a planar domain and
observe the first passage points at which $N$ Brownian paths $(\gamma_1,...,
\gamma_N)$ first enter each chamber, under the condition that the loop-erased
parts $(\LE(\gamma_1),..., \LE(\gamma_N))$ make a system of nonintersecting
LEBPs in the domain in the sense of Fomin. We prove that the correlation
functions of first passage points of the Brownian paths of the present system
are generally given by determinants specified by a continuous function called
the correlation kernel. The correlation kernel is of Eynard-Mehta type, which
has appeared in two-matrix models and time-dependent matrix models studied in
random matrix theory. Conformal covariance of correlation functions is
demonstrated.
