Ryan O’Donnell

1. Explicit near-Ramanujan graphsof every degree Ryan O’Donnell Sidhanth Mohanty (Berkeley) Pedro Paredes (Carnegie Mellon) Carnegie Mellon

2. Overture

3. d-regular, n-vertexgraph G, with adj. matrix A Fix d, n, and any vertex v. Let T ⟶ ∞. even The number of closed walks v vv length T dT is Θ(______). Trace Method:ρ(A) = max |λi(A)| = the base of this exponential

4. If G is directed (so A is not symmetric) you look at AT/2 (A*)T/2… …closed walks where thefirst T/2 steps are forward,last T/2 steps are backward

5. d-regular graph G The number of closed walks ee length T/2 length T/2 (d−1)T is Θ(). Trace Method ⇒ ρ(B) = d−1 Look at nonbacktracking walks. Let B be the associated adj. matrix(rows/cols indexed by directed edges)

6. d-regular infinite tree The number of closednonbacktracking walks ee length T/2 length T/2 (d−1)T/2 is Θ(). ∴ ρ(B∞) = Key: The closed walks must reduplicate each edge. Image credit: Hoory−Linial−Wigderson

7. “But I care about A, not B!” Ihara−Bass Formula: For a d-regular graph G, is an eigenvalue of A β is an eigenvalue of B ⟺ (plus B has m−neigs of ±1) finite d-reggraphmax eigenvalue: d−1 d infinited-reg treemax eigenvalue:

8. On to the main topic

9. d-regular, n-vertexgraph G, with adj. matrix A Write A’s eigenvalues as d =λ1≥λ2≥ ··· ≥λn λ = max |λj| j≠1 Smaller λ = better expander

10. Smaller λ = better expander “Gn is an expander (family)” ⟺ λ≤ .99 d

11. Smaller λ = better expander (1−Ωn(1)) d “Gn is an expander (family)” ⟺ λ≤ [Alon−Boppana’86]: − on(1) λ≥ E.g.: There are only finitely many10-regular graphs with λ≤ 5.999.

12. Q: Are there infinitely many 10-regulargraphs Gn with λ≤6? 10 6? Yes, if 10−1=9 is a prime. Answer 1:[Iha’66,LPS’88,Mar’88] Ramanujan Graph:λ≤ [Morgenstern’94] Yes, if d−1 is a prime power. λ≥ They’re explicit: Gncomputable in deterministic poly(n) time. In fact,stronglyexplicit: G2ⁿ computable in deterministic in poly(n) time.

13. Q: Are there infinitely many 10-regulargraphs Gn with λ≤6? 7 ? Yes if you only want λ2≤ . Answer 2:[MSS’15] Their graphs are bipartite, hence have λn = −d. λ≥ [Cohen’16] These graphs can be made explicit.

14. Q: Are there infinitely many 10-regulargraphs Gn with λ≤6? 7 ? Answer 3:[Friedman’08,Bordenave’17] ∀ ϵ > 0, a randomd-regular graph Gn has λ≤ + ϵw.h.p. λ≥ Our result: We get explicit such Gn.

15. Explicit near-Ramanujan graphsof every degree Ryan O’Donnell Sidhanth Mohanty (Berkeley) Pedro Paredes (Carnegie Mellon) Carnegie Mellon

16. Q: Are there infinitely many 10-regulargraphs Gn with λ≤6? 7 ? Answer 3:[Friedman’08,Bordenave’17] ∀ ϵ > 0, a randomd-regular graph Gn has λ≤ + ϵw.h.p. λ≥ Our result: We get explicit such Gn. E.g.: In deterministic poly(n) time, we can construct an n-vertex, 101-regular graph G with λ≤20.000001.

17. “Probabilistically Strongly Explicit” [Bilu−Linial’06] det. poly(n)-time computable seed s∈{0,1}O(n) circuit Cn Cnimplements the adjacency list of G2ⁿ a 2n-vertex, d-regular graph W.h.p. over choice of s, we have λ≤ + ϵ (Implies explicit: replace n with logn; try all seeds; check.)

18. More prior work Zig-Zag Product:[RVW’02,BT’11] λ≤ , strongly explicit Iterated lifts:[BL’06] λ≤ , probabilisticallystrongly explicit Add matchings toLPS/Margulis:[CM’08] λ≤ , strongly explicit(assuming Cramér’s Conjecture)

19. About our work

20. So you want to derandomize this… ∀ ϵ > 0, a randomd-regular graph Gn has λ≤ + ϵw.h.p. [Friedman’08] [Bordenave’17] 100 pages 30 pages I understand about 70% of it. With 10% understanding, you’ll see thatO(log n)-wise independent permutationsderandomize it, in nO(log n)time.

21. So you want to derandomize this… ∀ ϵ > 0, a randomd-regular graph Gn has λ≤ + ϵw.h.p. With 10% understanding, you’ll see thatO(log n)-wise independent permutationsderandomize it, in nO(log n)time.

22. So you want to derandomize this… ∀ ϵ > 0, a randomd-regular graph Gn has λ≤ + ϵw.h.p. ∴ in deterministic poly(n) time, can construct d-regular Gn0 with λ ≤+ ϵ and also the “no bicycles” property

23. Our Main Technical Theorem Given any d-regular G0satisfying , a random 2-liftG2 has λ≤ max{ λ(G0), + ϵ} w.h.p. Facts: 1. G2is d-regular and |V(G2)|= 2·|V(G0)| 2. G2also satisfies 3. Easily derandomizable in poly(n) time using almost-(log n)-wise independent strings. [NN93]

24. weaklyderandomizingBordenave derand2-lift derand2-lift derand2-lift vertices in P • λ≤+ ϵ • λ≤+ ϵ • λ≤+ ϵ • λ≤+ ϵ in P in P in P • • • d-regular d-regular d-regular d-regular 4n0 vertices 2n0 vertices n vertices

25. End of derandomization On to random 2-lifts of fixed graphs

26. 2-lifts = edge-signings

27. 2-lifts = edge-signings + + + + + + + + + + + +

28. 2-lifts = edge-signings + + + + + + + + + + + +

29. 2-lifts = edge-signings + + + + + + + + + + + +

30. 2-lifts = edge-signings + + + + + + + + + + + +

31. 2-lifts = edge-signings + + + + + + + + + + − +

32. 2-lifts = edge-signings + + + + + + + + + + − +

33. 2-lifts = edge-signings + + + + + + + + + + − +

34. 2-lifts = edge-signings + + + + + + + + − + − +

35. 2-lifts = edge-signings + + + + + + + + − + − +

36. 2-lifts = edge-signings + + + + + + + + − + − + n-vertex 2n-vertex

37. 2-lifts = edge-signings + + + + + + + + − + − + d-regular d-regular

38. 2-lifts = edge-signings + + + + + + + + − + − + (similar to “girth ≥ L”)

39. 2-lifts = edge-signings G0,± G2 + + + contains d for sure,constructed to haveall other |eigs| ≤d+ϵ + + + hopefully has all|eigs| ≤ + ϵd + + − + − + Fact: eigs(G2) = eigs(G0) ∪ eigs(G±)

40. Our Main Technical Theorem Given any d-regular G0satisfying , a random 2-liftG2 has λ≤ max{ λ(G0), + ϵ} w.h.p. Facts: 1. G2is d-regular and |V(G2)|= 2·|V(G0)| 2. G2also satisfies 3. Easily derandomizable in poly(n) time using almost-(log n)-wise independent strings. [NN93]

41. Our Main Technical Theorem Given any d-regular G0satisfying , a random 2-liftG2 has λ≤ max{ λ(G0), + ϵ} w.h.p. Need to show: a random edge-signingG± has ρ(G±) ≤ + ϵw.h.p.

42. Our Actual Main Technical Theorem Given any d-regular G0satisfying , a random edge-signingG± has ρ(G±) ≤ + ϵw.h.p.

43. Our Actual Main Technical Theorem Given any d-regular G0satisfying , a random edge-signingG± has ρ(G±) ≤ + ϵw.h.p. Remark: Easy to show a random d-regular G super-satisfies w.h.p.

44. Hence our theorem implies: A random d-regular G with random edge-signs has ρ(G) ≤ + ϵw.h.p. • [Bordenave’17] would have proven this had he been asked • Implicit in [DMOSS’19,MOP’19] • If you want to see proof, I humbly suggest our paper • That will give you about 70% understanding of [Bordenave’17]

45. Our Main Technical Theorem Given any d-regular n-vertex G satisfying , a random edge-signingG± has ρ(G±) ≤ + ϵw.h.p. What is ?! Define L = log log n ·poly log log log n = ∀ v, the L-neighborhood of v has ≤1 cycle

46. v

47. Our Main Technical Theorem Given any d-regular n-vertex G satisfying , a random edge-signingG± has ρ(G±) ≤ + ϵw.h.p. What is ?! Define L = log log n ·poly log log log n = ∀ v, the L-neighborhood of v has ≤1 cycle Ex: Random G has this w.h.p., even for any L = o(log n)

48. weaklyderandomizingBordenave derand2-lift derand2-lift derand2-lift vertices in P • λ≤+ ϵ • λ≤+ ϵ • λ≤+ ϵ • λ≤+ ϵ in P in P in P • • • d-regular d-regular d-regular d-regular 4n0 vertices 2n0 vertices n vertices

49. weaklyderandomizingBordenave derand2-lift derand2-lift derand2-lift vertices • • • at radius ≈ 2n0 vertices 4n0 vertices n vertices

50. Our Main Technical Theorem Let G be d-regular, n-vertex, every log log n-nbhd has ≤1 cycle. Then a random edge-signing G± has ρ(G±) ≤ + ϵw.h.p.