Percolation Systems away from the Critical Point Deepak Dhar

This article reviews some effects of disorder in percolation systems even away from the critical density p c. For densities below p c , the statistics of large clusters defines the animals problem. Its relation to the directed animals problem and the Lee-Yang edge singularity problem is described. Rare compact clusters give rise to Griffiths singuraties in the free energy of diluted ferromagnets, and lead to a very slow relaxation of magnetization. In biassed diffusion on percolation clusters, trapping in dead-end branches leads to asymptotic drift velocity becoming zero for strong bias, and very slow relaxation of velocity near the critical bias field. 1 Introduction Let me start by thanking Professor H. R. Krishnamurthy and other members of the organizing committee for inviting me to this meeting to felicitate Professor Narendra Kumar on his 60th birthday, and giving me an opportunity to pay my tribute to him. Over years, I have always enjoyed discussing various questions with Narendra. The large spectrum of his interests, and his spirit of enquiry, and insights to disentangle the essential problem from confusing camouflage have been a source of admiration for me. Thus I am really very happy to come here, and express my respect for him, and join the other speakers in wishing him many more years of happy questing. While I have shared with Kumar a common interest in understanding disordered systems, my own work has been largely in classical statistical mechanics (¯ h = 0), while Kumar's contributions to quantum problems have been discussed by several speakers here. Even so, I can legitimately claim to be one of Kumar's coworkers. Our only paper together was on the behavior of ±J spin-glass on a Bethe lattice, and was presented at the DAE Solid State Physics Symposium meeting at Madras (now Chennai) in December, 1979. It turned out that neither Narendra nor I could attend the meeting, so the paper was presented by our host,