  Extensive numerical simulations in the past decades proved that the critical
exponents of the jamming of frictionless spherical particles remain unchanged
in two and three dimensions. This implies that the upper critical dimension is
$d_u=2$ or lower. In this work, we study the jamming transition below the upper
critical dimension. We investigate a quasi-one-dimensional system: disks
confined in a narrow channel. We show that the system is isostatic at the
jamming transition point as in the case of standard jamming transition of the
bulk systems in two and three dimensions. Nevertheless, the scaling of the
excess contact number shows the linear scaling. Furthermore, the gap
distribution remains finite even at the jamming transition point. These results
are qualitatively different from those of the bulk systems in two and three
dimensions.
