  We analyze ground-state properties of a large gated quantum dot coupled via a
quantum point contact to a reservoir of one-dimensional interacting spinless
electrons. We find that the classical step-like dependence of the dot
population on the gate voltage is preserved under certain conditions. We point
out that the problem is related to the classical one-dimensional Ising model
with inverse-square interactions. This Ising universality class further
subdivides into (i) the Kondo/Ising class and (ii) the tricritical class. For
systems of the Kondo/Ising class, and repulsive electrons, the gate voltage
dependence of the dot population is continuous for sufficiently open dots,
while taking the form of a modified staircase for dots sufficiently isolated
from the reservoir. At the phase transition between the two regimes the
magnitude of the dot population jump is universal. For systems in the
tricritical class we find in addition an intermediate regime where the dot
population jumps from near integer value to a region of stable population
centered about a half-integer value. In particular, this tricritical behaviour
(together with the two dependencies already seen in the Kondo/Ising class) is
realized for non-interacting electrons.
