I am getting unreasonably too small p-values from limma in my analysis. See the following minimal example in which a random variable has similar distributions in two conditions. So we expect to get an insignificant p-value. However, this is not the limma's output.

n1 <- 100
topTable(eBayes(lmFit(c(1:n1, 1:n1), c(rep(0,n1),rep(1,n1)))), num=Inf, coef=1)

The p-value in the output is very significant:

logFC AveExpr        t      P.Value    adj.P.Val        B
1  50.5    50.5 10.97056 3.320846e-22 3.320846e-22 12.81377

I am using limma Version 3.40.6.

sessionInfo()
R version 3.6.1 (2019-07-05)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Mojave 10.14.6

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] limma_3.40.6

loaded via a namespace (and not attached):
[1] compiler_3.6.1

Did you notice that your logFC and average expression are identical? That's because the coefficient you are testing is the mean of one of the groups, and you are testing to see if the average of that group is different from zero (which it is, hence the exceedingly small p-value). You should probably spend more time reading the limma User's Guide, and maybe Julian Faraway's linear models book.

Thanks James for the references. When I add an intercept column to the design matrix, the p-values are not significant anymore as expected:

topTable(eBayes(lmFit(c(1:n1, 1:n1), cbind(1,c(rep(0,n1),rep(1,n1))))), num=Inf, coef=2)
logFC AveExpr        t P.Value adj.P.Val    B
1 3.97e-16     5.5 2.93e-16       1         1 -4.6

I still need to figure out how the design matrix and intercept determine what is being tested. It seems that with intercept, we are comparing the averages between groups, and without intercept we are testing whether average of groups are zero.