1. AIM, Palo Alto, August 2006 T. Witten, University of Chicago

2. Elements of a phase transition a set of N variables, e.g. "spins" { s i } = 0 or 1: any particular choice is a configuration a list or graph of connections between variables e.g. square lattice with neighbors connected statistical weight for each configuration eg "each connected pair of unlike spins reduces weight by factor x" a rule for joining joining N systems by adding connections eg. enlarge lattice. Such a system may have a phase transition: averages change non-smoothly with weights eg d s dx x = x c N s x 0 1 Non-smoothness requires N ––emergent Phase 1Phase 2 Variables are influenced by an infinite number of others 0 1 11 1 1 11 11 1 1 10 00 0 0 0 0 0 0 0 0 01

3. Equilibrium phase transitions –T. Witten, University of Chicago varieties of phase transition nature of the transition state theoretical methods questions for workshop distinctions: two non-phase transitions Mean field theory Conformal symmetry Renormalization theory Stochastic Loewner evolution Complex emergent structure with mysterious regularities

4. Phase transition versus ? s x 0 1 non abrupt: crossover non-emergent: abrupt without N

5. Ising model: paradigm phase transition Probabilities P{s} are set by a "goal function" H{s} Goal: all s's are the same as their connected neighbors Specifically: H{s} = -J (s c1 - 1/2)(s c2 - 1/2) connections c=1 L 0 1 11 1 1 11 11 1 1 10 00 0 0 0 0 0 0 0 0 01 Unlike neighbors increased H decreased probability P: P{s} = (constant) e -J/2 (= x) if s c1 s c2 1 if s c1 = s c2` connections c Has a phase transition with discontinuous derivitive d s dx … at x = x c N (and L)

6. Generalizing Ising model Variables: Z 2 (Ising), Z n (clock), O 2 (superfluid helium) O n (magnets) … In general the variables are group elements and the goal function depends on the group operation relating the connected elements Graph: d-dimensional lattice, continuous space, Cayley tree … connections beyond nearest-neighbor connections among k > 2 variables Goal function: seeks like values of variables / seeks different values Satisfiable in finite fraction, vanishing fraction (Ising), or none (frustrated) of the configurations Probabilities: thermal equilibrium (Ising): each graph element contributes one factor to the probability; factor depends on goal function on that element. kinetic: configurations are generated by a sequential, stochastic process whose probabilities are dictated by a goal function Stochastic growth processes Definite / random: eg, each connection has weight x or 1/x chosen randomly: spin glass

7. "nearly empty lattice" generalizations Self-avoiding random walk Random animals percolation Goal function allows only sites that form a linear sequence from the origin. Penalty factor x for each site Goal function allows only cluster of connected sites Penalty x for each site Place s variables on a lattice at random with a given s = x Determine largest connected cluster All have phase transitions: as x x c, size of cluster grows to a nonzero fraction of the lattice

8. Varieties of phase-transition behavior How non-smooth? The transition state discontinuous continuous heterogeneous State depends on history, boundary conditions Jochen Voss: http://seehuhn.de/mathe/ising-0.219.jpg slow: correlation time diverges at x c complicated Discontinuous: expectation values eg s jump at transition point eg boiling Continuous: only derivitives of s are discontinuous: eg Ising model, critical opalescence aka first-order, subcritical aka second-, third- …order, critical Uniform regions: eg liquid and vapor System is uniform over length scales > "correlation length" x x c

9. Character of heterogeneity: dilation invariance Dilated configurations are indistinguishable Eg random walks with steps much smaller than resolution 4 of these are zoomed 3x relative to the other four. Can you tell which? s(0) s( r 1 )s( r 2 )…s( r k ) = -Ak s(0) s(r 1 )s(r 2 )…s(r k ) Dilation invariance means that statistical averages are virtually unchanged by dilation: A is the "scaling dimension" of s This invariance appears to hold generally for continuous phase transitions A governs response of length to a change of x ~ (x - x c ) -1/(d-A) A set of "critical exponents" like A dictate response near the transition point

10. Character of the transition state: universality "Relevant" features do change the exponents: Discrete universality classes Spatial dimension d Critical exponents like A are often invariant under continuous changes of the system Eg Ising model on different lattices, different variables, different goal functions Eg liquid vapor, ferromagnet, liquid demixing, ising These changes are called "irrelevant variables" Symmetry of the system: Z 2 (Ising), O 2, … O 3

11. Theoretical understanding of phase transitions Mean-field theory Renormalization theory Conformal symmetry Stochastic Loewner evolution (Defects (local departures from goal state) ….omitted) (replica methods, omitted)

12. Mean-field approximation: neglect nonuniformity In H(s), replace spins connected to s I by their average s This converts system to a single degree of freedom s I One can readily compute eg thermodynamic free energy F(x, s ) for assumed s The actual s is that which minimizes free energy.

13. Mean field theory accounts for phase transitions. 0.20.40.60.81 -0.68 -0.66 -0.64 -0.62 -0.6 -0.58 J =.45J =.55 free energy at temperature = 1 magnetization M energy p p entropy Inherently independent of space dimension d Average s "magnetization" M takes the value that minimizes free energy

14. Renormalization theory To explain dilation symmetry requires a dilation-symmetric description To describe large-scale structure, we may remove small-scale structure of {s} by averaging over distance >> graph connections to produce s (r) If H [s ] describes the system and s is dilation symmetric, then H 2 must have the same functional form as H Eg for d-dimensional Ising model with d 4 at transition H ( s) 2 + g*(d) s 4 One infers the mapping H H 2 from the weights P[ s 2 ]: H P[ s ] P[ s 2 ] H 2 Local spatial averaging: 2 From the transformation of H near transition point, one infers the effect of dilation: "scaling dimension" A of s other critical exponents. ––Wilson, Fisher 1972 Many H's converge to same H : explains universality

15. Conformal symmetry for 2-dimensional systems configuration in z plane g(z) shows deformed configuration Any analytic function g(z) Conformal symmetry: deformed configuration has same statistical weight as original. –Shankar, Friedan ~1980 Critical exponents label a discrete set of representations of conformal group Cf discrete angular momentum representations of rotation symmetry Restricted to d=2

16. Loewner: analytic map implements self-avoidance Cf. John Cardy cond-mat/0503313 Z g(z) = a + ((z-a) 2 + 4t) 1/2 g aa t) Any self-avoiding curve has a g(z), analytic in Growing curve: tip z 0 came from some point a t on the real axis. z0z0 By varying a t with time, we can make arbitrary self avoiding curves. Loewner: affect of a t on g t (z) is local in time: Note: if a t is held fixed for t, z 0 diffuses upward with diffusion constant 4 Each point on the current curve g t (z) feels Coulomb repulsion from a t.

17. Schramm: Loewner growth with Brownian a t suppose a t diffuses, with ( a t ) 2 = t Curve is dilation symmetric (fractal) with fractal dimension D = 1 - /8 Many known random lattice structures are Schramm curves, differing only in Self-avoiding walk: = 8/3 Ising cluster perimeters: = 3 Percolation cluster boundaries: = 6 … Brownian motion replaces field theory to explain many known universal dilation- symmetric structures. … simple random walks can create a new range of correlated objects.

18. Generalizing the notion of phase transitions Processes in this workshop (eg k-sat) act like phase transitions Sharp change of behavior Emergent structure Scale invariance complexity Without the features thought fundamental in conventional phase transition Statistical mechanics description Spatial dimensionality. Can we free phase transitions from these features to get a deeper understanding?

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21. Correlation qualitatively alters structure Random walk: no correlation Self-avoiding walk: correlation Size ~ length 1/2 Size ~ length 3/4 central limit theorem Strong modifications needed to defeat central limit theorem. …interacting field theory 3/4 exponent was only known by simulation conjectures

22. Future hopes for Schramm-Loewner (SLE) Can it be generalized to account for branched objects, eg. Random animals Can it describe stochastic growth phenomena, eg. DLA? Can it be generalized to higher dimensions? SLE shows that many known dilation-symmetric structures are simple kinetic, random walks, viewed through the distorting lens of evolving maps Random structures like self avoiding walks are universal fractals in any dimension If they are Schramm-like in two-dimensions, why not in others? What plays the role of g t (z)? Displacement field of mapping a region into itself Must preserve topology Must preserve dilation symmetry. Must have a singular point: growing tip.