| Summary: | We use an <i>m</i>-vicinity method to examine Ising models on hypercube lattices of high dimensions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>≥</mo><mn>3</mn></mrow></semantics></math></inline-formula>. This method is applicable for both short-range and long-range interactions. We introduce a small parameter, which determines whether the method can be used when calculating the free energy. When we account for interaction with the nearest neighbors only, the value of this parameter depends on the dimension of the lattice <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>d</mi></semantics></math></inline-formula>. We obtain an expression for the critical temperature in terms of the interaction constants that is in a good agreement with the results of computer simulations. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>=</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>7</mn></mrow></semantics></math></inline-formula>, our theoretical estimates match the numerical results both qualitatively and quantitatively. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>=</mo><mn>3</mn><mo>,</mo><mn>4</mn></mrow></semantics></math></inline-formula>, our method is sufficiently accurate for the calculation of the critical temperatures; however, it predicts a finite jump of the heat capacity at the critical point. In the case of the three-dimensional lattice (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>=</mo><mn>3</mn></mrow></semantics></math></inline-formula>), this contradicts the commonly accepted ideas of the type of the singularity at the critical point. For the four-dimensional lattice (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>=</mo><mn>4</mn></mrow></semantics></math></inline-formula>), the character of the singularity is under current discussion. For the dimensions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>2</mn></mrow></semantics></math></inline-formula> the <i>m</i>-vicinity method is not applicable.
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