  We investigate critical phenomena of the Yang-Mills (YM) type one-dimensional
matrix model that is a large-$N$ reduction (or dimensional reduction) of the
$D+1$ dimensional $U(N)$ pure YM theory (bosonic BFSS model). This model shows
a large-$N$ phase transition at finite temperature, which is analogous to the
confinement/deconfinement transition of the original YM theory. We study the
matrix model at a three-loop calculation via the "principle of minimum
sensitivity" and find that there is a critical dimension $D=35.5$: At $D \le
35$, the transition is of first order, while it is of second order at $D\ge
36$. Furthermore, we evaluate several observables in our method, and they
nicely reproduce the existing Monte Carlo results. Through the gauge/gravity
correspondence, the transition is expected to be related to a Gregory-Laflamme
transition in gravity, and we argue that the existence of the critical
dimension is qualitatively consistent with it. Besides, in the first order
transition case, a stable phase having negative specific heat appears in the
microcanonical ensemble, which is similar to Schwarzschild black holes. We
study some properties of this phase.

High Energy Physics - Theory;General Relativity and Quantum Cosmology;High Energy Physics - Lattice