  A new path integral representation of Lorentzian Engle-Pereira-Rovelli-Livine
(EPRL) spinfoam model is derived by employing the theory of unitary
representation of SL(2,$\mathbb{C}$). The path integral representation is taken
as a starting point of semiclassical analysis. The relation between the
spinfoam model and classical simplicial geometry is studied via the large spin
asymptotic expansion of the spinfoam amplitude with all spins uniformaly large.
More precisely in the large spin regime, there is an equivalence between the
spinfoam critical configuration (with certain nondegeneracy assumption) and a
classical Lorentzian simplicial geometry. Such an equivalence relation allows
us to classify the spinfoam critical configurations by their geometrical
interpretations, via two types of solution-generating maps. The equivalence
between spinfoam critical configuration and simplical geometry also allows us
to define the notion of globally oriented and time-oriented spinfoam critical
configuration. It is shown that only at the globally oriented and time-oriented
spinfoam critical configuration, the leading order contribution of spinfoam
large spin asymptotics gives precisely an exponential of Lorentzian Regge
action of General Relativity. At all other (unphysical) critical
configurations, spinfoam large spin asymptotics modifies the Regge action at
the leading order approximation.
