Fractal Geometry of Minimal Spanning Trees

Transcription

1 Fractal Geometry of Minimal Spanning Trees T. S. Jackson and N. Read Part I: arxiv: Part II: arxiv: Accepted by Phys. Rev. E

2 Problem Definition Given a graph with costs on the edges, the MST is A tree (no loops) Spanning: all vertices on the tree Minimum: the spanning tree which minimizes total cost of edges on the tree (= energy)

3 Problem Definition Given a graph with costs on the edges, the MST is A tree (no loops) Spanning: all vertices on the tree Minimum: the spanning tree which minimizes total cost of edges on the tree (= energy)

4 Problem Definition Given a graph with costs on the edges, the MST is A tree (no loops) Spanning: all vertices on the tree Minimum: the spanning tree which minimizes total cost of edges on the tree (= energy)

5 Relevant properties 1 4 Invariance under reparameterization of costs MST = Union of max. cost edge (barrier)- minimizing paths (geodesics) e.g.,,, Construction via greedy algorithms (no metastability, no RSB)

6 Motivation MST paths = solutions to strong disorder transport problems Strong disorder = extensive variance E.g., resistor networks, polymers Nontrivial results (exponents) for a model with quenched disorder Equivalently: nonlocal optimization problem in terms of local field theory Newman and Stein 94: mapped EA spin glass ground state in strong disorder limit to MSTs

7 Objects of study D of MST paths via path vertex connectedness function D of MSF clusters visible in window dc, via hyperscaling breakdown ( SD spin glass ground state enumeration)

8 Connection to Percolation Disorder: edge costs are IID quenched random variables uniformly distributed on [0,1] Kruskal s greedy algorithm for MST: add cheapest remaining edge unless a loop forms MST p 0 Bond percolation: raise occupation probability, always accept cheapest remaining edge

9 Rest of Talk Part I: Mean field theory on Bethe lattice MFT breaks down at dc = 6 How? MSF cluster proliferation from multiple percolation spanning clusters (e.g., Aizenman 97) Part II: Loop corrections for d < dc MSF cluster D = d, so look at MST paths Adapt field theoretic perturbation series for percolation Prove series is renormalizable to all orders 1-loop example RG calculation

10 Part I: Mean field theory No finite cycles

11 Bethe Lattice percolation Coordination z = σ + 1 = 3

12 Bethe Lattice percolation Coordination z = σ + 1 = 3 Percolation: F (p) = 1 p + pf σ (p) = + F (p) p c = 1/σ p

13 MST: Bethe Lattice MSTs

14 Bethe Lattice MSTs MST: Use wired boundary conditions

15 Bethe Lattice MSTs MST: Use wired boundary conditions MSF in bulk

16 Bethe Lattice MSTs MST: Use wired boundary conditions y MSF in bulk Q: Are two points connected in the bulk or via boundary? x

17 Bethe Lattice MSTs MST: Use wired boundary conditions y MSF in bulk Q: Are two points connected in the bulk or via boundary? x = Mean-field D of MSF clusters

18 Bethe Lattice MSTs MST: Use wired boundary conditions y MSF in bulk Q: Are two points connected in the bulk or via boundary? x = Mean-field D of MSF clusters = dc where hyperscaling, MFT break down

19 MFT connectedness functions Definitions: 0 j 1 j m P (0;m) = Pr = 0 m Pr Boundary

20 MFT connectedness functions Definitions: 0 j 1 j m P (0;m) = = 0 m l j 1,j l 0, l j 1, l j, l m, Boundary

21 MFT connectedness functions Definitions: 0 j 1 j m P (0;m) = = 0 m Work with rates Recursion: Φ j (p) = F (p) σ 1 (pφ j 1 (p) + Either Φ j (p) = d dp P (0;j) (p) Boundary p 0 j 1 j 0 j 1 j or 0 ) dp Φ j 1 (p ) = 1 0 dp [ F (p) σ 1 d dp[ pθ(p p ) ]] Φ j 1 (p ) K(p, p )

22 MFT connectedness functions Connectedness functions via iterated kernel: K m (p, p ) = Then and 1 0 dp 1 P (0;m) (p 0 ) = 1 = real-space propagator C (2) (m; p 0 ) = p0 0 0 m m =0 dp m 1 K(p, p 1 ) K(p m 1, p ) Initial condition: Φ 0 (p) = d dp (1 F (p)σ ) = Φ 0 (p) 0 & similarly for n-point functions = tree-level diagrams 0 m dp dp Φ 0 (p )K m (p, p )F (p) K m 0 K m m Φ 0;0,0 F F m m m

23 Kernel asymptotics Generating function discrete Fourier-Laplace transform Φ w (p) = j=0 w j Φ j (p) Invert via Green s function gw(p,p ) 1 0 = momentum space propagator ] dp 2 [δ(p 1 p 2 ) wk(p 1, p 2 ) g w (p 2, p 3 ) = δ(p 1 p 3 ) K m (p 1, p 2 ) = 1 2πi dw g w(p 1, p 2 ) w m+1 Extract large-m behavior from first pole encountered at wc 0 w c 1 w c p 2 F σ 1 (p 2 ) = 0

24 MFT scaling from lattice 1-pt. : P (0,m) = P (p 0 ) (σ + 1)σ m 1 Compare generating function w/fourier: m ~ r 2 Paths on MSF look like random walks Asymptotic regime: 2-pt.: MSF clusters have fractal dimension D = 6 dc = 6 for MSTs (k+1)-pt.: m sites C (2) (m; p 0 ) 1 3 (σδp 0) 2 m 2 σ m m sites m O ( (p 0 p c ) 1) C (k+1) = M(m) k m 3k P (0,m )(p 0 ) = mp (p 0 ) m sites correlation length C (2) (m ; p 0 ) m 3 δp 2 0 not multifractal

25 Part II: Kruskal clusters at d<dc, p pc Only finite cycles

26 Diagrammatics Percolation = q 1 Potts model; want geometric interpretation Return to low-p expansion for percolation on any finite graph (Essam 72, 80) Percolation: two points connected if there exists at least one path between them Expand using inclusion-exclusion on paths Group all subsets of paths w/same support sum over subgraphs Equivalent form using subsets of edges Graphs carry associated diagrammatic weight identical to q 1 Potts tensor contractions

27 Diagrammatics MSF problem: want Pr[x,y connected by path through z] Can identify MST path between two points as minimum of ordering relation Consequence of geodesic property ultrametricity (transport applications) Weight now depends on barrier cost ordering π Inclusion-exclusion still applicable; must sum over orderings of each graph C z x,y(l 0 ) = G G x,y,z Pr[G l 0 ] π S E(G) d MSF (G π) Pr[π], (IV.6)

28 Example calculation Percolation (3-pt) d(g) = ( 1) E E I [E connects x, y, z] E E z +1-6 x y +11 d(g) = 4-2

29 Example calculation d MSF (G π) = E E MSF path vertex ( 1) E E I[E connects x, y] I[γ MST (G E π E ) passes through z] 0 x z d MSF (G π) = 1 y

30 Example calculation d MSF (G π) = E E MSF path vertex ( 1) E E I[E connects x, y] I[γ MST (G E π E ) passes through z] +1 x z d MSF (G π ) = 0 y

31 Lattice to continuum Sum over graphs of same topology continuum expansion d(g), dmsf(g π) invariant under insertion of paths of edges Proper weight for orderings? Only need keep max. cost edge (barrier) along any path Pr[L 1 < L 2 p 0 ] = Toy Example: p0 0 Pr[G p 0 ] = Generalization straightforward... L 1 N 2 e E(G) N 1 L 2 p 0 = p N 1 0 pn 2 0 p2 d d dp 2 dp 1 P G (p 1, p 2 ) 0 dp 1 dp 2 N 2 = p N 1+N 2 0 N 1 + N 2

32 Continuum theory Generating functions of random walk = massive propagator for scalar field Can neglect excluded volume constraint for RG Interpret mass on each propagator as MST barrier for given path Higher barrier p formed later as Kruskal is run smaller value of t O MSF (π, t 0 ) = >t π(1) t π(2) t π(n) t 0 ɛ E dt ɛ d dt ɛ acting on Feynman diagram of φ 3 theory

33 Continuum theory Generating functions of random walk = massive propagator for scalar field Can neglect excluded volume constraint for RG Interpret mass on each propagator as MST barrier for given path Higher barrier p formed later as Kruskal is run smaller value of t O MSF (π, t 0 ) = >t π(1) t π(2) t π(n) t 0 ɛ E dt ɛ Diagrammatic weights unchanged d dt ɛ acting on Feynman diagram of φ 3 theory

34 To summarize To generate the Feynman graph expansion for the MSF path vertex function, Start with graphs for the mass insertion in a massive cubic theory; For each graph and each ordering of edge masses, compute dmsf(g π) for that ordering, assign each edge a unique mass and act with OMSF(t0) for that ordering and sum over orderings and perform integration over loop momenta of Feynman diagram. C z x,y(p 0 ) G π S E d MSF (G π )O MSF (π, t 0 )I(G, {t ɛ })

35 Renormalizability The (possible) problem: Γ P V =

36 Renormalizability The (possible) problem: Γ P V = Need contribution from any subgraph to be independent of costs outside

37 Renormalizability The (possible) problem: Γ P V = Need contribution from any subgraph to be independent of costs outside Not true for lattice MST or expansion!

38 Renormalizability The (possible) problem: Γ P V = Need contribution from any subgraph to be independent of costs outside Not true for lattice MST or expansion! RG simplifications:

39 Renormalizability The (possible) problem: Γ P V = Need contribution from any subgraph to be independent of costs outside Not true for lattice MST or expansion! RG simplifications: Only keep UV-divergent terms = positive powers of Λ

40 Renormalizability The (possible) problem: Γ P V = Need contribution from any subgraph to be independent of costs outside Not true for lattice MST or expansion! RG simplifications: Only keep UV-divergent terms = positive powers of Λ z x H y 2- and 3-pt. subgraphs only x G y

41 Renormalizability The (possible) problem: Γ P V = Need contribution from any subgraph to be independent of costs outside Not true for lattice MST or expansion! RG simplifications: Only keep UV-divergent terms = positive powers of Λ 2- and 3-pt. subgraphs only x z G/H y

42 Renormalizability OMSF factorizes Only UV-divergent contributions to any diagram come from a subset of all orderings Short paths form first High-momentum propagators get higher masses dmsf factorizes for these orderings (not in general) Two-point diagram gives barrier mass insertion

43 d MSF (RGrelevant) Factorization of dmsf ( z ) g 1 g 2 g z g n x y Percolation{ (1PI) d MSF d MSF ( z ) x H ( z ) g z n = d MSF d x y y ( z ) x H = y ( ) = d H ( ) d H i=1 d MSF d MSF ( ) g i ( z ) x y ( z ) x y Path vertex: ( z ) ( z ) x x ( ) x d MSF H = d MSF H d y y MSF z y Barrier: d MSF ( z ) x H y = ( ) d H d MSF ( t H z ) x y

44 1 loop RG q d 2 = 1 d 3 = β(u 0 ) = ε 2 u d MSF ( t 1, t 2 > t 3 ) = 0 ( ) d2 4 d 3 else = 1 u O(u 5 0, εu 3 0) d PV = 2 3 γ φ (u 0 ) = d 2 6 u2 0 + O(u 4 0, εu 2 0) ( γ P V (u 0 ) = d P V d ) 2 u O(u 4 6 0, εu 2 0) (ε = d c d) D MSF = 2 (6d P V d 2 ) 3(d 2 4d 3 ) ε = 2 ε 7 + O(ε2 ) singly connected bonds: D perc = ν 1 perc = 2 5ε 21 + O(ε2 )

45 Future directions Behavior for p > pc Application to other problems? Loop-erased random walk from self-avoiding walk Two-replica formulation of ±J EA spin glass Metastable states??? Ground state enumeration? Perturbation away from strong disorder Failure of greedy algorithms related to, e.g., RSB instability?

46 Summary MST problem has dc = 6, not 8 D MST = 2 ε 7 + O(ε2 ) Treatment of quenched disorder in strong limit without (explicit?) replicas, cavity method, FRG, supersymmetry,... Nonlocality perturbation series not generated by (local) action Simplification within RG framework Part I: TSJ and N. Read, arxiv: Part II: TSJ and N. Read, arxiv: