Let X be a compact normal complex space of dimension n and L be a holomorphic line bundle on X. Suppose that \(\Sigma =(\Sigma _1,\ldots ,\Sigma _\ell )\) is an \(\ell \) -tuple of distinct irreducible proper analytic subsets of X, \(\tau =(\tau _1,\ldots ,\tau _\ell )\) is an \(\ell \) -tuple of positive real numbers, and let \(H^0_0(X,L^p)\) be the space of holomorphic sections of \(L^p:=L^{\otimes p}\) that vanish to order at least \(\tau _jp\) along \(\Sigma _j\) , \(1\le j\le \ell \) . If \(Y\subset X\) is an irreducible analytic subset of dimension m, we consider the space \(H^0_0 (X|Y, L^p)\) of holomorphic sections of \(L^p|_Y\) that extend to global holomorphic sections in \(H^0_0(X,L^p)\) . Assuming that the triplet \((L,\Sigma ,\tau )\) is big in the sense that \(\dim H^0_0(X,L^p)\sim p^n\) , we give a general condition on Y to ensure that \(\dim H^0_0(X|Y,L^p)\sim p^m\) . When L is endowed with a continuous Hermitian metric, we show that the Fubini-Study currents of the spaces \(H^0_0(X|Y,L^p)\) converge to a certain equilibrium current on Y. We apply this to the study of the equidistribution of zeros in Y of random holomorphic sections in \(H^0_0(X|Y,L^p)\) as \(p\rightarrow \infty \) .
