Multiparameter persistence modules can be uniquely decomposed into indecomposable summands. Among these indecomposables, intervals stand out for their simplicity, making them preferable for their ease of interpretation in practical applications and their computational efficiency. Empirical observations indicate that modules that decompose into only intervals are rare. To support this observation, we show that for numerous common multiparameter constructions, such as density- or degree-Rips bifiltrations, and across a general category of point samples, the probability of the homology-induced persistence module decomposing into intervals goes to zero as the sample size goes to infinity.

翻译：多参数持久性模可被唯一分解为不可约直和项。在这些不可约分量中，区间模因其简单性而尤为突出，这使它们在实践应用中更易于解释且计算效率更高。经验观察表明，仅由区间模分解而成的模块十分罕见。为支持这一观察结论，我们证明：对于密度型或度数型Rips双过滤等常见多参数构造，以及一般类别的采样点样本，当样本量趋于无穷时，由同调诱导的持久性模分解为区间模的概率趋近于零。