Expander graphs – applications and combinatorial constructions Avi Wigderson IAS, Princeton [Hoory, Linial, W. 2006] “Expander graphs and applications” Bulletin of the AMS.

Applications in Math & CS

Applications of Expanders In CS Derandomization Circuit Complexity Error Correcting Codes Data Structures … Computational Information Approximate Counting Communication & Sorting Networks

Applications of Expanders In Pure Math Graph Theory - … Group Theory – generating random group elements [Babai,Lubotzky-Pak] Number Theory Thin Sets [Ajtai-Iwaniec-Komlos-Pintz- Szemeredi] -Sieve method [Bourgain-Gamburd-Sarnak] - Distribution of integer points on spheres [Venkatesh] Measure Theory – Ruziewicz Problem [Drinfeld, Lubotzky-Phillips-Sarnak], F-spaces [Kalton-Rogers] Topology – expanding manifolds [Brooks] - Baum-Connes Conjecture [Gromov]

Expander graphs: Definition and basic properties

Expanding Graphs - Properties Combinatorial: no small cuts, high connectivity Probabilistic: rapid convergence of random walk Algebraic: small second eigenvalue Theorem. [Cheeger, Buser, Tanner, Alon-Milman, Alon, Jerrum-Sinclair,…]: All properties are equivalent!

Expanding Graphs - Properties Combinatorial: no small cuts, high connectivity Geometric: high isoperimetry G(V,E) V vertices, E edges |V|=n (  ∞ ) d-regular (d fixed) d  S |S|< n/2 |E(S,S c )| > α |S|d (what we expect in a random graph) α constant S

Expanding Graphs - Properties Probabilistic: rapid convergence of random walk G(V,E) d-regular v 1, v 2, v 3,…, v t,… v k+1 a random neighbor of v k v t converges to the uniform distribution in O(log n) steps (as fast as possible)

Expanding Graphs - Properties Algebraic: small second eigenvalue G(V,E) V V A G A G (u,v) = normalized adjacency matrix (random walk matrix) 0 (u,v)  E 1/d (u,v)  E 1 = 1 ≥ 2 ≥ … ≥ n ≥ -1 (G) = max i>1 | i | = max {  A G v  :  v  =1, v  u } (G) ≤  < 1 1- (G) “spectral gap”

Expanders - Definition Undirected, regular (multi)graphs. G is [n,d]-graph: n vertices, d-regular. Theorem: An [n,d]-graph G is connected iff (G) <1. Indeed, if G is (non-bip) connected than (G) <1- 1/dn 2 G is [n,d,  ]-graph: (G)  . G expander if  <1. Definition: An infinite family {G i } of [n i,d,  ]-graphs is an expander family if for all i  <1.

Random walk convergence (algebraic exp.  fast mixing) G(V,E) V V A G A G (u,v) = 0 (u,v)  E 1/d (u,v)  E 1 = 1 ≥ 2 ≥ … ≥ n eigenvalues u = v 1 v 2 … v n eigenvectors p: any probability distribution on V p (t) = (A G ) t p: distribution after t steps of a random walk  p (t) – u  1   n  p (t) – u  =  n  (A G ) t (p–u)    n ( (G)) t Converges in O(log n) steps. Corollary: Diam (G) < O(log n).

Expander mixing lemma [Alon-Chung] (algebraic exp.  combinatorial exp.) G(V,E) V V A G A G (u,v) = 0 (u,v)  E 1/d (u,v)  E 1 = 1 ≥ 2 ≥ … ≥ n eigenvalues u = v 1 v 2 … v n eigenvectors S  V 1 S =  i  i v i  1 = |S|/  n T  V 1 T =  i  i v i  1 = |T|/  n  1 S A G 1 T  =  i i  i  i = |S||T|/n +  i>1 i  i  I ||E(S,T)|- d |S||T|/n |  d (G)  i>1  i  i  d (G)  |S||T|  d (G)n Expected cut size: like a random graph A G  J/n

Existence, explicit construction and basic parameters

Existence of sparse expanders Theorem [Pinsker] Most 3-regular graphs are expanders. Proof: The probabilistic method (an early application). Take a random 3-regular graph G(V,E) with |V|=n. Fix a set S  V, |S| = s < n/2. Pr[|E(S,S c )| <.1s] < (s/n) 3s Pr[ G is not expanding ] <  s ( ) (s/n) 3s < 1/2 Challenge: Explicit (small degree) expanders! nsns

Explicit construction G(V,E) V V A G A G (u,v) = 0 (u,v)  E 1/d (u,v)  E {G i (V i,E i )} infinite sequence of graphs Weakly explicit: A G i can be constructed in poly (|V i |) time Strongly explicit: A G i can be constructed in polylog (|V i |) time: A poly(n) algorithm, given i, u  V i, lists all v  V i s.t. (u,v)  E i

How small can (G) be? Amplification: Consider G k G is [n,d,  ]-graph (with  <1) then G k is [n,d k,  k ]-graph So increasing d we can make (G) = 1/d c for some c>0 Theorem: [Alon-Boppana] An infinite family {G i } of [n i,d]-graphs must have (G) > (2  ( d-1))/d  1/  d Proof: (of a weaker statement). Assume d < n/2. n/d = Tr[A G A G t ] =  i i 2 < 1+(n-1) (G) 2  1/  2 d < (G) V V A G

Basic consequences: G [n,d,  ]- graph Theorem. If S,T  V, (u,v)  E a random edge, then | Pr[u  S and v  T] -  (S)  (T) | <  Cor 1: A random neighbor of a random vertex in S will land in S with probability <  (S)   Cor 2: Every set of size >  n contains an edge.  Chromatic number (G) > 1/   Graphs of large girth and chromatic number Cor 3: Removing any fraction  <  of the edges leaves a connected component of 1-O(  ) of the vertices.

Fault-tolerant computation

Infection Processes: G [n,d,  ]- graph,  <1/4 Cor 4: Every set S of size s 2  s neighbors in S Infection process 1: Adversary infects I 0, | I 0 |   n /4. I 0 =S 0, S 1, S 2, …S t,… are defined by: v  S t+1 iff a majority of its neighbors are in S t. Fact: S t =  for t > log n [infection dies out] Proof: |S i+1 |≤|S i |/2 Infection process 2: Adversary picks I 0, I 1,…, | I t |   n /4. I 0 =R 0, R 1, R 2, …R t,… are defined by R t = S t  I t Fact: |R t |   n /2 for all t [infection never spreads]

Reliable circuits from unreliable components [von Neumann] X2X2 X3X3 f V V V V V X1X1 V Given, a circuit C for f of size s Every gate fails with prob p < 1/10 Construct C’ for C’(x)=f(x) whp. Possible? With small s’?

Reliable circuits from unreliable components [von Neumann] X2X2 X3X3 f V V V V V X1X1 V Given, a circuit C for f of size s Every gate fails with prob p < 1/10 Construct C’ for C’(x)=f(x) whp. Possible? With small s’? - Add Identity gates I I I I I I

1 Reliable circuits from unreliable components [von Neumann] f Given, a circuit C for f of size s Every gate fails with prob p < 1/10 Construct C’ for C’(x)=f(x) whp. Possible? With small s’? -Add Identity gates -Replicate circuit -Reduce errors X2X2 X3X3 V V V V V X1X1 V I I I I I I X2X2 X3X3 V V V V V X1X1 V I I I I I I X2X2 X3X3 V V V V V X1X1 V I I I I I I X2X2 X3X3 V V V V V X1X1 V I I I I I I

Reliable circuits from unreliable components [von Neumann, Dobrushin-Ortyukov, Pippenger] Given, a circuit C for f of size s Every gate fails with prob p < 1/10 Construct C’ for C’(x)=f(x) whp. Possible? With small s’? Majority “expanders” of size O(log s)  Analysis: Infection Process 2 1 f X2X2 X3X3 V V V V V X1X1 V M X2X2 X3X3 V V V V V X1X1 V M X2X2 X3X3 V V V V V X1X1 V M X2X2 X3X3 V V V V V X1X1 V M MMMM MMMM MMMMMMMMMMMM

Bipartite expanders

(unbalanced) Bipartite expanders G(V,E) V’ V’’ u w u’w’ u”w” V’’’ H(V’,V’’’;E*) |V’|=n, |V’’’|= (2/3) n, degree=4d  S  V’ |S|  n/2, |N V’’’ (S)|  |S|  S has a perfect matching to V’’’  S  V’ |S|  n/2, |N V’’ (S)|  (3/2) |S|

Concentrators C n [Bassalygo,Pinsker] C n  S  V’ |S|  n/2, S has a perfect matching to V’’ V’ V’’ n 2n/3

Networks - Fault-tolerance - Routing - Distributed computing - Sorting

Superconcentrators [Valiant] | I|=|O|=n  k  n SC n  I’  I  O’  O |I’|=|O’|=k There are k vertex disjoint paths I’ to O’ I O How many edges are needed for SC n ? [V] This number is a Circuit lower bound for Disc. Fourier Transform

Superconcentrators & circuits [Valiant] [V] |E(SC n )| is a circuit lower bound for linear circuits computing any matrix A with all minors nonsingular y = Ax  k  n  I’  I  O’  O |I’|=|O’|=k there are k Vertex disjoint paths I’ to O’ Otherwise, by Menger’s Theorem,  (k-1)-cut, so Rank(A I’,O’ ) <k I O X 1 X 2 X 3 X n Y 1 Y 2 Y 3 Y n       

Superconcentrators [Valiant] Theorem: [Valiant] Linear size SC n Proof: [Pippenger] (1)|I’|=|O’|  n/2 recursion (2) |I’|=|O’|> n/2 Matching & pigeonhole Reduce to case (1) |E(SC n )|= = |E(SC 2n/3 )|+2|E(C n )|+n = |E(SC 2n/3 )|+O(n) = O(n) I O SC n SC 2n/3 CnCn CnCn MnMn

Distributed routing [Sh,PY,Up,ALM,AC…] G n inputs, n outputs, many disjoint paths - Permutation networks - Non-blocking network - Wide-sense non-blocking networks - on-line routable networks - …. Building block G - some expander Theorem: [Alon-Capalbo] Let G be a sufficiently strong expander. Given (s 1,t 1 ),(s 2,t 2 ),…(s k,t k ), k < n/(log n), one can efficiently find (s i,t i ) edge-disjoint paths between them, on-line!

Sorting networks [Ajtai-Komlos-Szemeredi] n inputs (real numbers), n outputs (sorted) Many sorting algorithms of O(n log n) comparisons Many sorting networks of O(n log 2 n) comparators Thm: [AKS] Explicit network with O(n log n) comparators Proof: Extremely sophisticated use & analysis of expanders Cor: Monotone Boolean formula for majority (derandomizing a probabilistic existence proof of Valiant)

Derandomization

Deterministic error reduction Algxx rr {0,1} n random strings Thm [Chernoff] r 1 r 2 …. r k independent (kn random bits) Thm [AKS] r 1 r 2 …. r k random path (n+ O(k) random bits) Algxx rkrk xx r1r1 Majority G [2 n,d, 1/8 ]-graph G explicit! BxBx Pr[error] < 1/3 then Pr[error] = Pr[|{r 1 r 2 …. r k }  B x }| > k/2] < exp(-k) |B x |<2 n /3

Metric embeddings

Metric embeddings (into l 2 ) Def: A metric space (X,d) embeds with distortion  into l 2 if  f : X  l 2 such that for all x,y d(x,y)   f(x)-f(y)    d(x,y) Theorem: [Bourgain] Every n-point metric space has a O(log n) embedding into l 2 Theorem: [Linial-London-Rabinovich] This is tight! Let (X,d) be the distance metric of an [n,d]-expander G. Proof:  f,(A G -J/n)f   (G)  f  2 ( 2ab = a 2 +b 2 -(a-b) 2 ) (1- (G) ) E x,y [(f(x)-f(y)) 2 ]  E x~y [(f(x)-f(y)) 2 ] (Poincare inequality) (clog n) 2   22 All pairsNeighbors

Metric embeddings (into l 2 ) Def: A metric space (X,d) has a coarse embedding into l 2 if  f : X  l 2 and increasing, unbounded functions ,  :R  R such that for all x,y  ( d(x,y))   f(x)-f(y)  2   ( d(x,y)) Theorem: [Gromov] There exists a finitely generated, finitely presented group, whose Cayley graph metric has no coarse embedding into l 2 Proof: Uses an infinite sequence of Cayley expanders… Comment: Relevant to the Novikov & Baum-Connes conjectures

Nonlinear spectral gaps and metric embeddings into convex spaces Poincare inequality: G any expander.    f : V  l 2 E x,y [  f(x)-f(y)  2 ]   E x~y [  f(x)-f(y)  2 ]  = 1/(1- (G) ) Theorem: [Matousek] G any expander.   p  f : V  l p E x,y [  f(x)-f(y)  ]   p E x~y [  f(x)-f(y)  ] Theorem: [Lafforgue, Mendel-Naor] Construct explicit G (super-expander)  K   K  f : V  l p E x,y [  f(x)-f(y)  ]   K E x~y [  f(x)-f(y)  ] Theorem: [Kasparov-Yu, Gromov] Such family of constant degree expanders give rise to a metric space X with no coarse embedding into any uniformly convex space. pppp pPpP 2K2K 2K2K

Constructions

Expansion of Finite Groups G finite group, S  G, symmetric. The Cayley graph Cay(G;S) has x  sx for all x  G, s  S. Cay(C n ; {-1,1}) Cay(F 2 n ; {e 1,e 2,…,e n }) (G)  1-1/n 2 (G)  1-1/n Basic Q: for which G,S is Cay(G;S) expanding ?

Algebraic explicit constructions [Margulis,Gaber- Galil,Alon-Milman,Lubotzky-Philips-Sarnak,…Nikolov,Kassabov,..] Theorem. [LPS] Cay(A,S) is an expander family. Proof: “The mother group approach”: Appeals to a property of SL 2 (Z) [Selberg’s 3/16 thm] Strongly explicit: Say that we need n bits to describe a matrix M in SL 2 (p). |V|=exp(n) Computing the 4 neighbors of M requires poly(n) time! A = SL 2 (p) : group 2 x 2 matrices of det 1 over Z p. S = { M 1, M 2 } : M 1 = ( ), M 2 = ( )

Algebraic Constructions (cont.) Very explicit -- computing neighbourhoods in logspace Gives optimal results G n family of [n,d]-graphs -- Theorem. [AB] d (G n )  2  (d-1) --Theorem. [LPS,M] Explicit d (G n )  2  (d-1) (Ramanujan graphs) Recent results: -- Theorem [KLN] All* finite simple groups expand. -- Theorem [H,BG] SL 2 (p) expands with most generators. -- Theorem [BGT] same for all Chevalley groups

Zigzag graph product Combinatorial construction of expanders

Explicit Constructions (Combinatorial) -Zigzag Product [Reingold-Vadhan-W] G an [n, m,  ]-graph. H an [m, d,  ]-graph. Combinatorial construction of expanders. H v-cloud Edges Step in cloud Step between clouds Step In cloud v u u-cloud (v,k)(v,k) Thm. [RVW] G z H is an [ nm,d 2,  +  ]-graph, Definition. G z H has vertices {(v,k) : v  G, k  H}. G z H is an expander iff G and H are.

Proof of the zigzag theorem Proof: Information theoretic view of expanders. When is G an expander? Random walk is entropy boost! p, p’ distributions on V before and after a random step. “Def”:G is an expander iff whenever Ent(v) << log n, Ent(v’) > Ent(v) +  (  >0 ) Ent 0 (p) = log |supp(p)| Ent 1 (p) = Shannon’s ent. Ent 2 (p) = log  p  2 p v v’ p’

Proof of the zigzag thm [Reingold-Vadhan-W] H v-cloud vu u-cloud Thm. [RVW] G, H expanders, then so is G z H (u,c)(u,c) (u,d)(u,d) Want: Ent(u,d) > Ent(v,a) +  (assuming Ent(v,a) << log nm) Case 1: Ent(a|v) Ent(a|v) +   Ent(u,d)  Ent(v,b)  Ent(v,a) +  Case 2: Ent(a|v)= log m  Ent(b|v)= log m  Ent(v) << log n  Ent(u)  Ent(v) +   Ent(c|u) << log m  Ent(u,d)  Ent(u,c) +  (v,b)(v,b) (v,a)(v,a) Mutually exclusive? Linear Algebra!

Iterative Construction of Expanders G an [n,m,  ]-graph. H an [m,d,  ] -graph. The construction: Start with a constant size H a [d 4,d,1/4]-graph. G 1 = H 2 Theorem. [RVW] G k is a [d 4k, d 2, ½]-graph. Proof: G k 2 is a [d 4k,d 4, ¼]-graph. H is a [d 4, d, ¼]-graph. G k+1 is a [d 4(k+1), d 2, ½]-graph. Theorem. [RVW] G z H is an [nm,d 2,  +  ]-graph. G k+1 = G k 2 z H Weakly explicit construction. Strongly: ( G k  G k ) 2 z H

Consequences of the zigzag product - Isoperimetric inequalities beating e-value bounds [Reingold-Vadhan-W, Capalbo-Reingold-Vadhan-W] - Connection with semi-direct product in groups [Alon-Lubotzky-W] - New expanding Cayley graphs for non-simple groups [Meshulam-W] : Iterated group algebras [Rozenman-Shalev-W] : Iterated wreath products - SL=L : Escaping every maze deterministically [Reingold ’05] - Super-expanders [Mendel-Naor] - Monotone expanders [Dvir-W]

SL=L: escaping mazes and navigating unknown terrains

Getting out of mazes / Navigating unknown terrains (without map & memory) Theseus Ariadne Thm [Aleliunas-Karp- Lipton-Lovasz-Rackoff ’80] A random walk will visit every vertex in n 2 steps (with probability >99% ) Only a local view (logspace) n–vertex maze/graph Thm [Reingold ‘05] : SL=L: A deterministic walk, computable in Logspace, will visit every vertex. Uses ZigZag expanders Crete 1000BC Mars 2006

Expander from any connected graph [Reingold] Analogy with the iterative construction G an [n,m,  ]-graph. H an [m,d,  ] -graph. The construction: Fix a constant size H a [d 4,d, 1/4 ]-graph. G 1 = H 2 Theorem. [RVW] G k is a [d 4k, d 2, ½]-graph Proof: G k 2 is a [d 4k,d 4, ¼]-graph. H is a [d 4, d, ¼]-graph. G k+1 is a [d 4(k+1), d 2, ½]-graph. Theorem. G z H is an [nm,d 2,  +  ]-graph. G k+1 = G k 2 z H G an [n,m, 1-  ]-graph. H an [m,d, 1/4 ] -graph. [nm,d 2, 1-  /2 ]-graph. H a [d 10,d, 1/4 ]-graph. G 1 = G G k+1 = G k 5 z H G clog n is [n O(1), d 2, ½] Theorem [R] G 1 is [n, d 2, 1-1/n 3 ] 

Undirected connectivity in Logspace [ R ] Algorithm - Input G=G 1 an [n,d 2 ]-graph - Compute G clog n -Try all paths of length clog n from vertex 1. Correctness - G i+1 is connected iff G i is - If G is connected than it is an [n,d 2, 1-1/n 3 ]-graph - G 1 connected  G clog n has diameter < clog n Space bound - G i  G i+1 in constant space (squaring & zigzag are local) - G clog n from G 1 requires O(log n) space

Zigzag graph product & Semi-direct group product Expansion in non-simple groups

Semi-direct Product of groups A, B groups. B acts on A as automorphisms. Let a b denote the action of b on a. Definition. A  B has elements {(a,b) : a  A, b  B}. group mult (a’,b’ ) (a,b) = (a’a b, b’b) Connection: semi-direct product is a special case of zigzag Assume = B, = A, S = s B (S is a single B-orbit) Proof: By inspection (a,b)(1,t) = (a,bt) (Step in a cloud) (a,b)(s,1) = (as b,b) (Step between clouds) Theorem [ALW] Cay(A  B, TsT ) = Cay(A,S ) z Cay(B,T ) Theorem [ALW] Expansion is not a group property Theorem [MW,RSW] Iterative construction of Cayley expanders

Example: = A=F 2 m, the vector space, S={e1, e2,…, em}, the unit vectors B=Z m, the cyclic group, T={  1}, shift by 1 B acts on A by shifting coordinates. S=e 1 B. G =Cay(A,S), H = Cay(B,T), and G z H = Cay(A  B, {  1 }e 1 {  1 } ) z

Is expansion a group property? [Lubotzky-Weiss’93] Is there a group G, and two generating subsets |S 1 |,|S 2 |=O(1) such that Cay(G;S 1 ) expands but Cay(G;S 2 ) doesn’t ? (call such G schizophrenic) nonEx1: C n - no S expands nonEx2: SL 2 (p)- every S expands[Breuillard-Gamburd’09] [Alon-Lubotzky-W’01] SL 2 (p)  (F 2 ) p+1 schizophrenic [Kassabov’05] Sym n schizophrenic

Expansion in Near-Abelian Groups G group. [G;G] commutator subgroup of G [G;G] = G= G 0 > G 1 > … > G k = G k+1 G i+1 =[G i ;G i ] G is k-step solvable if G k =1. Abelian groups are 1-step solvable [Lubotzky-Weiss’93] If G is k-step solvable, Cay(G;S) expanding, then |S| ≥ O(log (k) |G|) [Meshulam-W’04] There exists k-step solvable G k, |S k | ≤ O(log (k/2) |G k |), and Cay(G k ;S k ) expanding. loglog….log k times

Near-constant degree expanders for near Abelian groups [Meshulam-W’04] Iterate: G’= G  F q G Start with G 1 = Z 2 Get G 1, G 2,…, G k,… |G k+1 |>exp (|G k |) S 1, S 2,…, S k,… = G k |S k+1 |<poly (|S k |) - Cay(G k, S k ) expanding - |S k |  O(log (k/2) |G k |) deg “approaching” constant Theorem/Conjecture: Cay(G,S) expands, then G has at most exp(d) irreducible reps of dimension d.

Expanders from iterated wreath products [Rozenman-Shalev-W’04] d fixed. G k = Aut*(T d k ) Odd automorphisms of a depth-k uniform d-ary tree Iterative: G k+1 = G k  A d Thm: G k expands with explicit O(1) generators Ingredients: zigzag, equations in perfect groups[Nikolov], correlated random walks expand,… 00 11 33 22 d=3, k=2  i  A 3

Beating eigenvalue expansion

Beating e-value expansion [WZ, RVW] In the following a is a large constant. Task: Construct an [n,d]-graph s.t. every two sets of size n/a are connected by an edge. Minimize d Ramanujan graphs: d=  (a 2 ) Random graphs: d=O(a log a) Zig-zag graphs: [RVW] d=O(a(log a) O(1) ) Uses zig-zag product on extractors! Applications Sorting & selection in rounds, Superconcentrators,…

Lossless expanders [Capalbo-Reingold-Vadhan-W] Task: Construct an [n,d]-graph in which every set S, |S| c|S| neighbors. Max c (vertex expansion) Upper bound: c  d Ramanujan graphs: [Kahale] c  d/2 Random graphs: c  (1-  )d Lossless Zig-zag graphs: [CRVW] c  (1-  )d Lossless Use zig-zag product on conductors! Extends to unbalanced bipartite graphs. Applications (where the factor of 2 matters): Data structures, Network routing, Error-correcting codes

Error correcting codes

Error Correcting Codes [Shannon, Hamming] C: {0,1} k  {0,1} n C=Im(C) Rate (C) = k/n Dist (C) = min d H (C(x),C(y)) C good if Rate (C) =  (1), Dist (C) =  (n) Theorem: [Shannon ‘48] Good codes exist (prob. method) Challenge: Find good, explicit, efficient codes. - Many explicit algebraic constructions: [Hamming, BCH, Reed-Solomon, Reed-muller, Goppa,…] - Combinatorial constructions [Gallager, Tanner, Luby- Mitzenmacher-Shokrollahi-Spielman, Sipser-Spielman..] Thm: [Spielman] good, explicit, O(n) encoding & decoding Inspiration: Superconcentrator construction!

Graph-based Codes [Gallager’60s] C: {0,1} k  {0,1} n C=Im(C) Rate (C) = k/n Dist (C) = min d H (C(x),C(y)) C good if Rate (C) =  (1), Dist (C) =  (n) z  C iff Pz=0 C is a linear code LDPC: Low Density Parity Check (G has constant degree) Trivial Rate (C)  k/n, Encoding time = O(n 2 ) G lossless  Dist (C) =  (n), Decoding time = O(n) n n-k z Pz G

Decoding Thm [CRVW] Can explicitly construct graphs: k=n/2, bottom deg = 10,  B  [n], |B|  n/200, |  (B)|  9|B| B = corrupted positions (|B|  n/200) B’ = set of corrupted positions after flip Decoding algorithm [Sipser-Spielman]: while Pw  0 flip all w i with i  FLIP = { i :  (i) has more 1’s than 0’s } Claim [SS] : |B’|  |B|/2 Proof: |B \ FLIP |  |B|/4, |FLIP \ B |  |B|/4 n n-k w Pw

Rapidly mixing Markov chains: Uniform generation and enumeration problems Expansion of (exponential-size) graphs which arise naturally (as opposed to specially designed)

Volumes of convex bodies Algorithm: [Dyer-Frieze-Kannan ‘91]: Approx counting  random sampling Random walk inside K. Rapidly mixing Markov chain. Techniques: Spectral gap  isoperimetric inequality Given (implicitly) a convex body K in R d (d large!) (e.g. by a set of linear inequalities) Estimate volume (K) Comment: Computing volume(K) exactly is #P-complete K

Statistical mechanics Example: the dimer problem Count # of domino configurations? Given G, count the number of perfect matchings G Glauber Dynamics (MCMC sampling the Gibbs distribution) Construct H G on configurations, with edges representing local changes (e.g. rotate adjacent parallel dominos). Then run a random walk for “sufficiently long” time. Theorems:[Jerrum-Sinclair] poly(n) time convergence for -near-uniform perfect matching in dense & random graphs -sampling Gibbs dist in ferromagnetic Ising model Techniques: coupling, conductance,… v  H G approximately

Generating random group elements Given: S={g 1,g 2,…g d } generators of a group G (of size n) Find: a near-uniform element x of G (  g  G: Pr[x=g] <  /n ) Straight-line program (SLP) [Babai-Szemeredi]: x 1,x 2,…x t where every x i is either: (1)A generator g j or g j -1 (2) x k x m or x m -1 for m,k < i Theorem:[BS]  G,S,g  an SLP for g of length t=O(log 2 n) Theorem:[Babai]  G,S  a near-uniform SLP* of t=O(log 5 n) Theorem:[Cooperman,Dixon,Green] …. SLP* of t=O(log 2 n) Proof: Cube(z 1,z 2,…z t ) = {subwords of z t -1 …z 2 -1 z 1 -1 z 1 z 2 …z t ) x 1,x 2,…x r with x k+1  R Cube(x 1,x 2,…x k ), with r=O(log n) Techniques: local expansion, Arithmetic combinatorics

Extensions of expanders

Dimension Expanders Expander Permutations  1,  2, …,  k : [n]  [n] are an expander if for every subset S  [n], |S| < n/2  i  [k] s.t. |T i S  S| < (1-  )|S| Dimension expander [Barak-Impagliazzo-Shpilka-W’01] Linear operators T 1,T 2, …,T k : F n  F n are (n,F)- dimension expander if subspace V  F n with dim(V) < n/2  i  [k] s.t. dim(T i V  V) < (1-  ) dim(V) Fact: k=O(1) random T i ’s suffice for every F,n. [Lubotzky-Zelmanov’04] Construction for F=C What about finite fields?

Monotone Expanders f: [n]  [n] partial monotone map: x<y and f(x),f(y) defined, then f(x)<f(y). f 1,f 2, …,f k : [n]  [n] are a k-monotone expander if f i partial monotone and the (undirected) graph on [n] with edges (x,f i (x)) for all x,i, is an expander. [Dvir-Shpilka] k-monotone exp  2k-dimension exp  F,d Explicit (log n)-monotone expander [Dvir-W’09] Explicit (log*n)-monotone expander (zig-zag) [Bourgain’09] Explicit O(1)-monotone expander [Dvir-W’09] Existence  Explicit reduction Open: Prove that O(1)-mon exp exist!

Real Monotone Expanders [Bourgain’09] Explicitly constructs f 1,f 2, …,f k : [0,1]  [0,1] continuous, Lipshitz, monotone maps, such that for every S  [0,1] with  (S)< ½, there exists i  [k] such that  (S  f 2 (S)) < (1-  )  (S) Monotone expanders on [n] – by discretization M=( )  SL 2 (R), x  R, let f M (x) = (ax+b)/(cx+d) Take  ’-net of such M i ’s in an  ’’-ball around I. Proof: -Expansion in SU(2) [Bourgain-Gamburd] -Tits alternative [Bruillard] -…-… a b c d

Open Problems

Characterize: Cayley expanders Construct: Lossless expanders Construct: Rate concentrators of constant left degree N 3 N2 N2 |X|  N |Γ(X)|  |X|