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We investigate critical phenomena of the Yang-Mills (YM) type one-dimensionalmatrix 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 showsa large-$N$ phase transition at finite temperature, which is analogous to theconfinement/deconfinement transition of the original YM theory. We study thematrix model at a three-loop calculation via the "principle of minimumsensitivity" and find that there is a critical dimension $D=35.5$: At $D \le35$, the transition is of first order, while it is of second order at $D\ge36$. Furthermore, we evaluate several observables in our method, and theynicely reproduce the existing Monte Carlo results. Through the gauge/gravitycorrespondence, the transition is expected to be related to a Gregory-Laflammetransition in gravity, and we argue that the existence of the criticaldimension is qualitatively consistent with it. Besides, in the first ordertransition case, a stable phase having negative specific heat appears in themicrocanonical ensemble, which is similar to Schwarzschild black holes. Westudy some properties of this phase.

Critical Dimension and Negative Specific Heat in One-dimensional Large-N Reduced Models