Tilings, Geometric Representations, and Discrete Analytic Functions László Lovász Microsoft Research

Harmonic functions Every non-constant function has at least 2 poles.

Example 1: Random walks S f(v)= E (f(Z v )) Z v : (random) point where random walk from v hits S v 0 v 1 f(v)= P ( random walk from v hits t before s) s t

Example 2: Electrical networks 0 v 1 f(v)= electrical potential s t

Example 3: Rubber bands f(v)= position of nodes 0 1

“Dictionary” of quantities from electrical networks, random walks, and static:

Analytic functions on the lattice (Ferrand 1944, Duffin 1956) z z+1 z+i z+i+1

“z”“z” i 2-i 1+i 1-i-i i 2i 3i 1+2i 4 3+i 3-i

i 3-4i 2i -2i i i 8-6i

i -1+i i -i-i -6i -19i -7-2i i -1-i2-7i i 1-i 1+i

Brooks-Smith-Stone-Tutte

Rotation-free circulation: Discrete holomorphic forms on surfaces Mercat 2001 Weighted version:

Grid on the torus

A : null-homologous circulationsdim( A ) = #faces-1= f-1 B : potentials dim( B ) = #nodes-1= n-1 C : rotation-free circulations dim( C ) = #edges- (#nodes-1)- (#faces-1) = 2·genus mutually orthogonal Discrete Hodge decomposition

“Analytic function cannot vanish on a large set’’ “Large” is not size: planar piece separating set of size < 2g

f : nonzero rotation-free circulation G ’: subgraph where f does not vanish U : connected subgraph such that f vanishes on all edges incident with U “Analytic function cannot vanish on a large set’’ U can be separated from G ’ by at most 4g-3 nodes. Benjamini-L

Main (easy) topological lemma:

(assume no other 0’s) Proof of Theorem u

Which sets of edges can be supports of rotation-free circulations? -  weighting of the edges - for weights =1 - for some weighting Can be characterized using matroid theory LL-Schrijver Combinatorially analytic maps:

incidence vector of edge uv projection of  uv on C If g>0 and G is 3-connected and simple, then  uv  0 for every edge uv. u v qp If g=0, then  uv =0 for every edge uv.

ee e

incidence vector of edge uv projection of  uv on C If g>0 and G is 3-connected and simple, then  uv  0 for every edge uv. u v qp If g>0 and G is 3-connected and simple, then  nowhere-0 rotation-free circulation on G.

A strange identity: If G is a 3-connected map, then

Toroidal maps: analytic functions, straight line embeddings, rubber bands, square tilings

Universal cover map

g=1 : Two linearly independent rotation-free circulations

Two linearly independent harmonic functions

Two coordinates  periodic embedding in the plane  embedding in the torus This gives an embedding in the torus

Horizontal coordinate is nowhere-zero flow  nondegenerate squares Edge  square horizontal length  size

R. Kenyon

Which other properties of analytic functions have discrete analogues? Multiplication of analytic (meromorphic) functions? Merkat: weight the edges, “critical” weighting Riemann-Roch? (Cai)

Global information from local observation “Can you hear the shape of a drum?” - Observe a graph locally (a single node, a neighborhood of a node, a “window”) - There is a local random process on the graph (random walk, heat bath,...) - Infer global properties of the graph

The graph, with the window:

The Noisy Circulator: (face balancing) (edge excitation) frequency 1 frequency p (node balancing)

If no edge excitation occurs, then the weighting converges to a rotation-free flow. dimension of rotation-free flows =2g dimension of rotation-free flows in window =2g if edge excitations occur infrequently: random vectors from a (2g) -dimensional subspace with errors

If p<N -c, then observing the Noisy Circulator for N c /p steps, we can determine g with high probability.