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Random Field Ising Model on Small-World Networks Seung Woo Son, Hawoong Jeong 1 and Jae Dong Noh 2 1 Dept. Physics, Korea Advanced Institute Science and Technology (KAIST) 2 Dept. Physics, Chungnam National University, Daejeon, KOREA

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2 What is RFIM ? ex) 2D square lattice Ising magnet Quenched Random Magnetic Field H i : Random Fields Ising Model cf) Diluted AntiFerromagnet in a Field (DAFF) Random field Uniform field

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3 RFIM on SW networks Ising magnet (spin) is on each node where quenched random fields are applied. Spin interacts with the nearest-neighbor spins which are connected by links. L : number of nodes K : number of out-going links p : random rewiring probability

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Why should we study this problem? Just curiosity + Critical phenomena in a stat. mech. system with quenched disorder. Applications : e.g., network effect in markets Individuals Society Tachy MSN Selection of an item = Ising spin state Preference to a specific item = random field on each node -Internet & telephone business -Messenger -IBM PC vs. Mac -Key board (QWERTY vs. Dvorak) -Video tape (VHS vs. Beta) -Cyworld ? Social science

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5 Zero temperature ( T=0 ) RFIM provides a basis for understanding the interplay between ordering and disorder induced by quenched impurities. Many studies indicate that the ordered phase is dominated by a zero-temperature fixed point. The ground state of RFIM can be found exactly using optimization algorithms (Max- flow, min-cut).

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6 Magnetic fields distribution Bimodal dist. Hat dist.

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7 Finite size scaling Finite size scaling form Limiting behavior ∆c∆c

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8 Binder cumulant Results on regular networks Hat distribution L (# of nodes) = 100 K (# of out-going edges of each node) = 5 P (rewiring probability) = 0.0

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9 Results on regular networks Hat distribution no phase transition

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10 Results on SW networks Hat distribution Binder cumulant L (# of nodes) = 100 K (# of out-going edges of each node) = 5 P (rewiring probability) = 0.5

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11 Results on SW networks Hat distribution

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12 Results on SW networks Second order phase transition Hat distribution

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13 Results on SW networks Bimodal distribution

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14 Results on SW networks First order phase transition Bimodal field dist.

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15 Summary We study the RFIM on SW networks at T=0 using exact optimization method. We calculate the magnetization and obtain the magnetization exponent(β) and correlation exponent (ν) from scaling relation. The results shows β/ν = 0.16, 1/ν = 0.4 under hat field distribution. From mean field theory β MF =1/2, ν MF =1/2 and upper critical dimension of RFIM is 6. ν* = d u v MF = 3 and β MF /ν* = 1/6, 1/ν* = 1/3. R. Botet et al, Phys. Rev. Lett. 49, 478 (1982).

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