  During the recent solar minimum between cycles 23 and 24 (solar minimum
$P_{23/24}$) the intensity of Galactic Cosmic Rays (GCRs) measured at the Earth
was the highest ever recorded since space age. It is the purpose of this paper
to resolve the most plausible mechanism for this unusually high intensity. A
GCR transport model in three-dimensional heliosphere based on a simulation of
Markov stochastic process is used to find the relation of cosmic ray modulation
to various transport parameters, including solar wind (SW) speed, distance of
heliospheric boundary, magnitude of interplanetary magnetic field (IMF) at the
Earth, tilt angle of heliospheric current sheet (HCS), values of parallel and
perpendicular diffusion coefficients. We calculate GCR proton energy spectra at
the Earth for the last three solar minima $P_{21/22}$, $P_{22/23}$, and
$P_{23/24}$, with the transport parameters obtained from observations. Besides
weak IMF magnitude and slow SW speed, we find that a possible low magnetic
turbulence, which increases the parallel diffusion and reduces the
perpendicular diffusion in the polar direction, might be an additional possible
mechanism for the high GCR intensity in the solar minimum $P_{23/24}$.
