Asymptotic behavior of Markov complexity of matrices

To any integer matrix $A$ one can associate a matroid structure consisting of a graph and another integer matrix $A_B$. The connected components of this graph are called bouquets. We prove that bouquets behave well with respect to the $r$–th Lawrence liftings of matrices and we use it to prove that the Markov and Graver complexities of $m\times n$ matrices of rank $d$ may be arbitrarily large for $n\ge 4$ and $d\le n-2$. In contrast, we show they are bounded in terms of $n$ and the largest absolute value $a$ of any entry of $A$.