  Define a chi-square random field on a multi-dimensional lattice points index
set with a direct-product covariance structure, and consider the distribution
of the maximum of this random field. We provide two approximate formulas for
the upper tail probability of the distribution based on nonlinear renewal
theory and an integral-geometric approach called the volume-of-tube method.
This study is motivated by the detection problem of the interactive loci pairs
which play an important role in forming biological species. The joint
distribution of scan statistics for detecting the pairs is regarded as the
chi-square random field above, and hence the multiplicity-adjusted $p$-value
can be calculated by using the proposed approximate formulas. By using these
formulas, we examine the data of Mizuta, et al. (2010) who reported a new
interactive loci pair of rice inter-subspecies.
