We describe non-conventional localization of the midband E=0 state in square and cubic finite bipartite lattices with off-diagonal disorder by solving numerically the linear equations for the corresponding amplitudes. This state is shown to display multifractal fluctuations, having many sparse peaks, and by scaling the participation ratio we obtain its disorder-dependent fractal dimension $D_{2}$. A logarithmic average correlation function grows as $g(r) sim eta ln r$ at distance $r$ from the maximum amplitude and is consistent with a typical overall power-law decay $|psi(r)| sim r^{-eta}$ where $eta $ is proportional to the strength of off-diagonal disorder.
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