  The probability of large deviations of the smallest Schmidt eigenvalue for
random pure states of bipartite systems, denoted as $A$ and $B$, is computed
analytically using a Coulomb gas method. It is shown that this probability, for
large $N$, goes as $\exp[-\beta N^2\Phi(\zeta)]$, where the parameter $\beta$
is the Dyson index of the ensemble, $\zeta$ is the large deviation parameter
while the rate function $\Phi(\zeta)$ is calculated exactly. Corresponding
equilibrium Coulomb charge density is derived for its large deviations. Effects
of the large deviations of the extreme (largest and smallest) Schmidt
eigenvalues on the bipartite entanglement are studied using the von Neumann
entropy. Effect of these deviations is also studied on the entanglement between
subsystems $1$ and $2$, obtained by further partitioning the subsystem $A$,
using the properties of the density matrix's partial transpose
$\rho_{12}^\Gamma$. The density of states of $\rho_{12}^\Gamma$ is found to be
close to the Wigner's semicircle law with these large deviations. The
entanglement properties are captured very well by a simple random matrix model
for the partial transpose. The model predicts the entanglement transition
across a critical large deviation parameter $\zeta$. Log negativity is used to
quantify the entanglement between subsystems $1$ and $2$. Analytical formulas
for it are derived using the simple model. Numerical simulations are in
excellent agreement with the analytical results.
