| Summary: | We study vacuum fluctuation properties of an ensemble of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>U</mi> <mo>(</mo> <mi>N</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> gauge theory configurations, in the limit of many colors, viz. <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>c</mi> </msub> <mo>→</mo> <mo>∞</mo> </mrow> </semantics> </math> </inline-formula>, and explore the statistical nature of the topological susceptibility by analyzing its critical behavior at a non-zero-vacuum parameter <inline-formula> <math display="inline"> <semantics> <mi>θ</mi> </semantics> </math> </inline-formula> and temperature <i>T</i>. We find that the system undergoes a vacuum phase transition at the chiral symmetry restoration temperature as well as at an absolute value of <inline-formula> <math display="inline"> <semantics> <mi>θ</mi> </semantics> </math> </inline-formula>. On the other hand, the long-range correlation length solely depends on <inline-formula> <math display="inline"> <semantics> <mi>θ</mi> </semantics> </math> </inline-formula> for the theories with critical exponent <inline-formula> <math display="inline"> <semantics> <mrow> <mi>e</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula> or <inline-formula> <math display="inline"> <semantics> <mrow> <mi>T</mi> <mo>=</mo> <msub> <mi>T</mi> <mi>d</mi> </msub> <mo>+</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <msub> <mi>T</mi> <mi>d</mi> </msub> </semantics> </math> </inline-formula> is the decoherence temperature. Furthermore, it is worth noticing that the unit-critical exponent vacuum configuration corresponds to a non-interacting statistical basis pertaining to a constant mass of <inline-formula> <math display="inline"> <semantics> <msup> <mi>η</mi> <mo>′</mo> </msup> </semantics> </math> </inline-formula>.
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