| Summary: | Tensor rank decomposition is a useful tool for geometric interpretation of the tensors in the canonical tensor model (CTM) of quantum gravity. In order to understand the stability of this interpretation, it is important to be able to estimate how many tensor rank decompositions can approximate a given tensor. More precisely, finding an approximate symmetric tensor rank decomposition of a symmetric tensor <i>Q</i> with an error allowance <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Δ</mi></semantics></math></inline-formula> is to find vectors <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>ϕ</mi><mi>i</mi></msup></semantics></math></inline-formula> satisfying <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>∥</mo><mi>Q</mi><mo>−</mo></mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>R</mi></msubsup><msup><mi>ϕ</mi><mi>i</mi></msup><mo>⊗</mo><msup><mi>ϕ</mi><mi>i</mi></msup><mo>⋯</mo><mo>⊗</mo><msup><mi>ϕ</mi><mi>i</mi></msup><msup><mrow><mo>∥</mo></mrow><mn>2</mn></msup><mo>≤</mo><mi mathvariant="sans-serif">Δ</mi></mrow></semantics></math></inline-formula>. The volume of all such possible <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>ϕ</mi><mi>i</mi></msup></semantics></math></inline-formula> is an interesting quantity which measures the amount of possible decompositions for a tensor <i>Q</i> within an allowance. While it would be difficult to evaluate this quantity for each <i>Q</i>, we find an explicit formula for a similar quantity by integrating over all <i>Q</i> of unit norm. The expression as a function of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Δ</mi></semantics></math></inline-formula> is given by the product of a hypergeometric function and a power function. By combining new numerical analysis and previous results, we conjecture a formula for the critical rank, yielding an estimate for the spacetime degrees of freedom of the CTM. We also extend the formula to generic decompositions of non-symmetric tensors in order to make our results more broadly applicable. Interestingly, the derivation depends on the existence (convergence) of the partition function of a matrix model which previously appeared in the context of the CTM.
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