Interlacing Families I: Bipartite Ramanujan Graphs of all Degrees - - PowerPoint PPT Presentation

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Interlacing Families I: Bipartite Ramanujan Graphs of all Degrees - - PowerPoint PPT Presentation

Jun 11, 2023 48 likes β€’1.31k views

Interlacing Families I: Bipartite Ramanujan Graphs of all Degrees Nikhil Srivastava (Microsoft Research) Adam Marcus (Yale), Daniel Spielman (Yale) Expander Graphs Sparse regular well-connected graphs with many properties of random graphs.

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SLIDE 1

Interlacing Families I: Bipartite Ramanujan Graphs of all Degrees

Nikhil Srivastava (Microsoft Research) Adam Marcus (Yale), Daniel Spielman (Yale)

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SLIDE 2

Expander Graphs

Sparse regular well-connected graphs with many properties of random graphs.

Every set of vertices has many neighbors. Random walks mix quickly. Pseudo-random generators. Error-correcting codes. Used throughout Computer Science.

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SLIDE 3

Spectral Expanders

Let G be a graph and A be its adjacency matrix eigenvalues πœ‡1 β‰₯ πœ‡2 β‰₯ β‹― πœ‡π‘œ

a c d e b

0 1 0 0 1 1 0 1 0 1 0 1 0 1 0 0 0 1 0 1 1 1 0 1 0

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SLIDE 4

Spectral Expanders

Let G be a graph and A be its adjacency matrix eigenvalues πœ‡1 β‰₯ πœ‡2 β‰₯ β‹― πœ‡π‘œ

If d-regular, then 𝐡𝟐 = π‘’πŸ so πœ‡1 = 𝑒 If bipartite then eigs are symmetric about zero so πœ‡π‘œ = βˆ’π‘’

a c d e b

0 1 0 0 1 1 0 1 0 1 0 1 0 1 0 0 0 1 0 1 1 1 0 1 0

β€œtrivial”

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SLIDE 5

Spectral Expanders

Definition: G is a good expander if all non-trivial eigenvalues are small

[ ]

  • d

d

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SLIDE 6

Spectral Expanders

Definition: G is a good expander if all non-trivial eigenvalues are small

[ ]

  • d

d e.g. 𝐿𝑒 and 𝐿𝑒,𝑒 have all nontrivial eigs 0.

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SLIDE 7

Spectral Expanders

Definition: G is a good expander if all non-trivial eigenvalues are small

[ ]

  • d

d e.g. 𝐿𝑒 and 𝐿𝑒,𝑒 have all nontrivial eigs 0.

Alon-Boppana’86: For every πœ— > 0, every sufficiently large d-regular graph has a nontrivial eigenvalue greater than2 𝑒 βˆ’ 1 βˆ’ πœ—

Challenge: construct infinite families.

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SLIDE 8

Ramanujan Graphs:

Definition: G is Ramanujan if all non-trivial eigs have absolute value at most 2 𝑒 βˆ’ 1

[ ]

  • d

d

[

  • 2 𝑒 βˆ’ 1

]

2 𝑒 βˆ’ 1

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SLIDE 9

Ramanujan Graphs:

Definition: G is Ramanujan if all non-trivial eigs have absolute value at most 2 𝑒 βˆ’ 1

[ ]

  • d

d

[

  • 2 𝑒 βˆ’ 1

]

2 𝑒 βˆ’ 1

Margulis, Lubotzky-Phillips-Sarnak’88: Infinite sequences of Ramanujan graphs exist for 𝑒 = π‘ž + 1 Friedman’08: A random d-regular graph is almost Ramanujan : 2 𝑒 βˆ’ 1 + πœ—

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SLIDE 10

Main Result

  • Theorem. Infinite families of bipartite Ramanujan graphs

exist for every 𝑒 β‰₯ 3.

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SLIDE 11

Main Result

  • Theorem. Infinite families of bipartite Ramanujan graphs

exist for every 𝑒 β‰₯ 3. Proof is elementary, doesn’t use number theory. Not explicit. Based on a new existence argument: method of interlacing families of polynomials.

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SLIDE 12

Bilu-Linial’06 Approach

Find an operation which doubles the size of a graph without blowing up its eigenvalues.

[ ]

  • d

d

[

  • 2 𝑒 βˆ’ 1

]

2 𝑒 βˆ’ 1

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SLIDE 13

Bilu-Linial’06 Approach

Find an operation which doubles the size of a graph without blowing up its eigenvalues.

[ ]

  • d

d

[

  • 2 𝑒 βˆ’ 1

]

2 𝑒 βˆ’ 1

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SLIDE 14

Bilu-Linial’06 Approach

Find an operation which doubles the size of a graph without blowing up its eigenvalues.

[ ]

  • d

d

[

  • 2 𝑒 βˆ’ 1

]

2 𝑒 βˆ’ 1

… ∞

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SLIDE 15

2-lifts of graphs

a c d e b

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SLIDE 16

2-lifts of graphs

a c d e b a c d e b

duplicate every vertex

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SLIDE 17

2-lifts of graphs

duplicate every vertex

a0 c0

d0

e0

b0

a1 c1

d1

e1

b1

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SLIDE 18

2-lifts of graphs

a0 c0

d0

e0

b0

a1

d1

e1

b1

c1

for every pair of edges: leave on either side (parallel),

  • r make both cross
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SLIDE 19

2-lifts of graphs

for every pair of edges: leave on either side (parallel),

  • r make both cross

a0 c0

d0

e0

b0

a1

d1

e1

b1

c1

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SLIDE 20

2-lifts of graphs

for every pair of edges: leave on either side (parallel),

  • r make both cross

a0 c0

d0

e0

b0

a1

d1

e1

b1

c1

2𝑛 possibilities

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SLIDE 21

2-lifts of graphs

0 1 0 0 1 1 0 1 0 1 0 1 0 1 0 0 0 1 0 1 1 1 0 1 0

n eigenvalues {πœ‡1 … πœ‡π‘œ}

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SLIDE 22

2-lifts of graphs

0 1 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 1 0

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SLIDE 23

2-lifts of graphs

0 0 0 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1

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SLIDE 24

2-lifts of graphs

0 0 0 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1

2n eigenvalues πœ‡1 … πœ‡π‘œ βˆͺ {πœ‡1

β€² … πœ‡π‘œ β€² }

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SLIDE 25

Eigenvalues of 2-lifts (Bilu-Linial)

Given a 2-lift of G, create a signed adjacency matrix As with a -1 for crossing edges and a 1 for parallel edges

0 -1 0 0 1

  • 1 0 1 0 1

0 1 0 -1 0 0 0 -1 0 1 1 1 0 1 0

a0 c0

d0

e0

b0

a1

d1

e1

b1

c1

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SLIDE 26

Eigenvalues of 2-lifts (Bilu-Linial)

Theorem: The eigenvalues of the 2-lift are the union of the eigenvalues of A (old) and the eigenvalues of As (new)

πœ‡1

β€² … πœ‡π‘œ β€²

= 𝑓𝑗𝑕𝑑(𝐡𝑑)

0 -1 0 0 1

  • 1 0 1 0 1

0 1 0 -1 0 0 0 -1 0 1 1 1 0 1 0

𝐡𝑑 =

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SLIDE 27

Eigenvalues of 2-lifts (Bilu-Linial)

Theorem: The eigenvalues of the 2-lift are the union of the eigenvalues of A (old) and the eigenvalues of As (new) Conjecture: Every d-regular graph has a 2-lift in which all the new eigenvalues have absolute value at most

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SLIDE 28

Eigenvalues of 2-lifts (Bilu-Linial)

Theorem: The eigenvalues of the 2-lift are the union of the eigenvalues of A (old) and the eigenvalues of As (new) Conjecture: Every d-regular adjacency matrix A has a signing 𝐡𝑑 with ||𝐡𝑇|| ≀ 2 𝑒 βˆ’ 1

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SLIDE 29

Eigenvalues of 2-lifts (Bilu-Linial)

Conjecture: Every d-regular adjacency matrix A has a signing 𝐡𝑑 with ||𝐡𝑇|| ≀ 2 𝑒 βˆ’ 1

We prove this in the bipartite case.

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SLIDE 30

Eigenvalues of 2-lifts (Bilu-Linial)

Theorem: Every d-regular adjacency matrix A has a signing 𝐡𝑑 with πœ‡1(𝐡𝑇) ≀ 2 𝑒 βˆ’ 1

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SLIDE 31

Eigenvalues of 2-lifts (Bilu-Linial)

Theorem: Every d-regular bipartite adjacency matrix A has a signing 𝐡𝑑 with ||𝐡𝑇|| ≀ 2 𝑒 βˆ’ 1 Trick: eigenvalues of bipartite graphs are symmetric about 0, so only need to bound largest

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SLIDE 32

Random Signings

Idea 1: Choose 𝑑 ∈ βˆ’1,1 𝑛 randomly.

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SLIDE 33

Random Signings

Idea 1: Choose 𝑑 ∈ βˆ’1,1 𝑛 randomly. Unfortunately, (Bilu-Linial showed when A is nearly Ramanujan )

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SLIDE 34

Random Signings

Idea 2: Observe that where

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SLIDE 35

Random Signings

Idea 2: Observe that where

Consider

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SLIDE 36

Random Signings

Idea 2: Observe that where Usually useless, but not here! is an interlacing family. such that

Consider

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SLIDE 37

3-Step Proof Strategy

  • 1. Show that some poly does as well as the .

such that

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SLIDE 38

3-Step Proof Strategy

  • 1. Show that some poly does as well as the .
  • 2. Calculate the expected polynomial.

such that

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SLIDE 39

3-Step Proof Strategy

  • 1. Show that some poly does as well as the .
  • 2. Calculate the expected polynomial.
  • 3. Bound the largest root of the expected poly.

such that

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SLIDE 40

3-Step Proof Strategy

  • 1. Show that some poly does as well as the .
  • 2. Calculate the expected polynomial.
  • 3. Bound the largest root of the expected poly.

such that

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SLIDE 41

3-Step Proof Strategy

  • 1. Show that some poly does as well as the .
  • 2. Calculate the expected polynomial.
  • 3. Bound the largest root of the expected poly.

such that

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SLIDE 42

Step 2: The expected polynomial

Theorem [Godsil-Gutman’81] For any graph G, the matching polynomial of G

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SLIDE 43

The matching polynomial (Heilmann-Lieb β€˜72)

mi = the number of matchings with i edges

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SLIDE 44
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SLIDE 45
  • ne matching with 0 edges
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SLIDE 46

7 matchings with 1 edge

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SLIDE 47
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SLIDE 48
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SLIDE 49

Proof that

x Β±1 0 0 Β±1 Β±1 Β±1 x Β±1 0 0 0 0 Β±1 x Β±1 0 0 0 Β±1 x Β±1 0 Β±1 0 0 Β±1 x Β±1 Β±1 0 0 0 Β±1 x

Expand using permutations

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SLIDE 50

Proof that

x Β±1 0 0 Β±1 Β±1 Β±1 x Β±1 0 0 0 0 Β±1 x Β±1 0 0 0 Β±1 x Β±1 0 Β±1 0 0 Β±1 x Β±1 Β±1 0 0 0 Β±1 x

Expand using permutations

same edge: same value

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SLIDE 51

Proof that

x Β±1 0 0 Β±1 Β±1 Β±1 x Β±1 0 0 0 0 Β±1 x Β±1 0 0 0 Β±1 x Β±1 0 Β±1 0 0 Β±1 x Β±1 Β±1 0 0 0 Β±1 x

Expand using permutations

same edge: same value

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SLIDE 52

Proof that

x Β±1 0 0 Β±1 Β±1 Β±1 x Β±1 0 0 0 0 Β±1 x Β±1 0 0 0 Β±1 x Β±1 0 Β±1 0 0 Β±1 x Β±1 Β±1 0 0 0 Β±1 x

Expand using permutations Get 0 if hit any 0s

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SLIDE 53

Proof that

x Β±1 0 0 Β±1 Β±1 Β±1 x Β±1 0 0 0 0 Β±1 x Β±1 0 0 0 Β±1 x Β±1 0 Β±1 0 0 Β±1 x Β±1 Β±1 0 0 0 Β±1 x

Expand using permutations Get 0 if take just one entry for any edge

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SLIDE 54

Proof that

x Β±1 0 0 Β±1 Β±1 Β±1 x Β±1 0 0 0 0 Β±1 x Β±1 0 0 0 Β±1 x Β±1 0 Β±1 0 0 Β±1 x Β±1 Β±1 0 0 0 Β±1 x

Expand using permutations Only permutations that count are involutions

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SLIDE 55

Proof that

x Β±1 0 0 Β±1 Β±1 Β±1 x Β±1 0 0 0 0 Β±1 x Β±1 0 0 0 Β±1 x Β±1 0 Β±1 0 0 Β±1 x Β±1 Β±1 0 0 0 Β±1 x

Expand using permutations Only permutations that count are involutions

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SLIDE 56

Proof that

x Β±1 0 0 Β±1 Β±1 Β±1 x Β±1 0 0 0 0 Β±1 x Β±1 0 0 0 Β±1 x Β±1 0 Β±1 0 0 Β±1 x Β±1 Β±1 0 0 0 Β±1 x

Expand using permutations Only permutations that count are involutions Correspond to matchings

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SLIDE 57

Proof that

x Β±1 0 0 Β±1 Β±1 Β±1 x Β±1 0 0 0 0 Β±1 x Β±1 0 0 0 Β±1 x Β±1 0 Β±1 0 0 Β±1 x Β±1 Β±1 0 0 0 Β±1 x

Expand using permutations Only permutations that count are involutions Correspond to matchings

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SLIDE 58

3-Step Proof Strategy

  • 1. Show that some poly does as well as the .
  • 2. Calculate the expected polynomial.

[Godsil-Gutman’81]

  • 3. Bound the largest root of the expected poly.

such that

οƒ–

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SLIDE 59

3-Step Proof Strategy

  • 1. Show that some poly does as well as the .
  • 2. Calculate the expected polynomial.

[Godsil-Gutman’81]

  • 3. Bound the largest root of the expected poly.

such that

οƒ–

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SLIDE 60

The matching polynomial (Heilmann-Lieb β€˜72)

Theorem (Heilmann-Lieb) all the roots are real

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SLIDE 61

The matching polynomial (Heilmann-Lieb β€˜72)

Theorem (Heilmann-Lieb) all the roots are real and have absolute value at most Proof: simple, based on recurrences.

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SLIDE 62

3-Step Proof Strategy

  • 1. Show that some poly does as well as the .
  • 2. Calculate the expected polynomial.

[Godsil-Gutman’81]

  • 3. Bound the largest root of the expected poly.

[Heilmann-Lieb’72] such that

οƒ– οƒ–

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SLIDE 63

3-Step Proof Strategy

  • 1. Show that some poly does as well as the .
  • 2. Calculate the expected polynomial.

[Godsil-Gutman’81]

  • 3. Bound the largest root of the expected poly.

[Heilmann-Lieb’72] such that

οƒ– οƒ–

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SLIDE 64

3-Step Proof Strategy

  • 1. Show that some poly does as well as the .

Implied by: β€œ is an interlacing family.” such that

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SLIDE 65

Averaging Polynomials

Basic Question: Given when are the roots

  • f the related to roots of ?
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SLIDE 66

Averaging Polynomials

Basic Question: Given when are the roots

  • f the related to roots of ?

Answer: Certainly not always

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SLIDE 67

Averaging Polynomials

Basic Question: Given when are the roots

  • f the related to roots of ?

Answer: Certainly not always…

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SLIDE 68

Averaging Polynomials

Basic Question: Given when are the roots

  • f the related to roots of ?

But sometimes it works:

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A Sufficient Condition

Basic Question: Given when are the roots

  • f the related to roots of ?

Answer: When they have a common interlacing.

  • Definition. interlaces

if

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SLIDE 70
  • Theorem. If have a common

interlacing,

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SLIDE 71
  • Theorem. If have a common

interlacing, Proof.

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  • Theorem. If have a common

interlacing, Proof.

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SLIDE 73
  • Theorem. If have a common

interlacing, Proof.

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SLIDE 74
  • Theorem. If have a common

interlacing, Proof.

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SLIDE 75
  • Theorem. If have a common

interlacing, Proof.

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SLIDE 76
  • Theorem. If have a common

interlacing, Proof.

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Interlacing Family of Polynomials

Definition: is an interlacing family

if can be placed on the leaves of a tree so that when every node is the sum of leaves below, sets of siblings have common interlacings

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SLIDE 78

Interlacing Family of Polynomials

Definition: is an interlacing family

if can be placed on the leaves of a tree so that when every node is the sum of leaves below, sets of siblings have common interlacings

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SLIDE 79

Interlacing Family of Polynomials

Definition: is an interlacing family

if can be placed on the leaves of a tree so that when every node is the sum of leaves below, sets of siblings have common interlacings

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SLIDE 80

Interlacing Family of Polynomials

Definition: is an interlacing family

if can be placed on the leaves of a tree so that when every node is the sum of leaves below, sets of siblings have common interlacings

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SLIDE 81

Interlacing Family of Polynomials

Definition: is an interlacing family

if can be placed on the leaves of a tree so that when every node is the sum of leaves below, sets of siblings have common interlacings

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SLIDE 82

Interlacing Family of Polynomials

Theorem: There is an s so that

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SLIDE 83

Proof: By common interlacing, one of , has

Interlacing Family of Polynomials

Theorem: There is an s so that

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SLIDE 84

Proof: By common interlacing, one of , has

Interlacing Family of Polynomials

Theorem: There is an s so that

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SLIDE 85

Proof: By common interlacing, one of , has

Interlacing Family of Polynomials

Theorem: There is an s so that

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SLIDE 86

Proof: By common interlacing, one of , has ….

Interlacing Family of Polynomials

Theorem: There is an s so that

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SLIDE 87

Proof: By common interlacing, one of , has

Interlacing Family of Polynomials

Theorem: There is an s so that

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SLIDE 88

An interlacing family

Theorem: Let is an interlacing family

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SLIDE 89

An interlacing family

Theorem: Let is an interlacing family Lemma (easy): and have a common interlacing if and only if is real rooted for all

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SLIDE 90

To prove interlacing family

Let Leaves of tree = signings 𝑑1, … , 𝑑𝑛 Internal nodes = partial signings 𝑑1, … , 𝑑𝑙

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SLIDE 91

To prove interlacing family

Let Need to prove that for all , is real rooted

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SLIDE 92

To prove interlacing family

Need to prove that for all ,

are fixed is 1 with probability , -1 with are uniformly

is real rooted Let

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SLIDE 93

Generalization of Heilmann-Lieb

Suffices to prove that is real rooted for every independent distribution

  • n the entries of s
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SLIDE 94

Generalization of Heilmann-Lieb

Suffices to prove that is real rooted for every independent distribution

  • n the entries of s:
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SLIDE 95

Transformation to PSD Matrices

Suffices to show real rootedness of

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SLIDE 96

Transformation to PSD Matrices

Suffices to show real rootedness of Why is this useful?

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SLIDE 97

Transformation to PSD Matrices

Suffices to show real rootedness of Why is this useful?

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SLIDE 98

Transformation to PSD Matrices

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SLIDE 99

π‘€π‘—π‘˜ = πœ€π‘— βˆ’ πœ€

π‘˜ with probability Ξ»π‘—π‘˜

πœ€π‘— + πœ€

π‘˜ with probability (1βˆ’πœ‡π‘—π‘˜)

Transformation to PSD Matrices

𝔽𝑑 det 𝑦𝐽 βˆ’ 𝑒𝐽 βˆ’ 𝐡𝑑 = 𝔽det 𝑦𝐽 βˆ’

π‘—π‘˜βˆˆπΉ

π‘€π‘—π‘˜π‘€π‘—π‘˜

π‘ˆ

where

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SLIDE 100

Master Real-Rootedness Theorem

Given any independent random vectors 𝑀1, … , 𝑀𝑛 ∈ ℝ𝑒, their expected characteristic polymomial has real roots.

𝔽det 𝑦𝐽 βˆ’

𝑗

𝑀𝑗𝑀𝑗

π‘ˆ

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SLIDE 101

Master Real-Rootedness Theorem

Given any independent random vectors 𝑀1, … , 𝑀𝑛 ∈ ℝ𝑒, their expected characteristic polymomial has real roots.

𝔽det 𝑦𝐽 βˆ’

𝑗

𝑀𝑗𝑀𝑗

π‘ˆ

How to prove this?

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SLIDE 102

The Multivariate Method

  • A. Sokal, 90’s-2005:

β€œβ€¦it is often useful to consider the multivariate polynomial … even if one is ultimately interested in a particular one-variable specialization”

Borcea-Branden 2007+: prove that univariate polynomials are real-rooted by showing that they are nice transformations of real-rooted multivariate polynomials.

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SLIDE 103

Real Stable Polynomials

is real stable if for all i Implies . Definition:

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SLIDE 104

Real Stable Polynomials

is real stable if no roots in the upper half-plane univariate real stable = real-rooted for all i Implies . Definition:

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SLIDE 105

If is real stable, then so is

  • 1. π‘ž(𝛽, 𝑨2, … , π‘¨π‘œ) for any 𝛽 ∈ ℝ
  • 2. 1 βˆ’ πœ–π‘¨π‘— π‘ž(𝑨1, … π‘¨π‘œ)

Excellent Closure Properties

is real stable if for all i Implies . Definition:

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SLIDE 106

A Useful Real Stable Poly

Borcea-BrΓ€ndΓ©n β€˜08: For PSD matrices is real stable

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SLIDE 107

A Useful Real Stable Poly

Borcea-BrΓ€ndΓ©n β€˜08: For PSD matrices is real stable Plan: apply closure properties to this to show that 𝔽det 𝑦𝐽 βˆ’ 𝑗 𝑀𝑗𝑀𝑗

π‘ˆ is real stable.

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SLIDE 108

Central Identity

Suppose 𝑀1, … , 𝑀𝑛 are independent random vectors with 𝐡𝑗 ≔ 𝔽𝑀𝑗𝑀𝑗

π‘ˆ. Then

𝔽det 𝑦𝐽 βˆ’

𝑗

𝑀𝑗𝑀𝑗

π‘ˆ

=

𝑗=1 𝑛

1 βˆ’ πœ– πœ–π‘¨π‘— det 𝑦𝐽 +

𝑗

𝑨𝑗𝐡𝑗

𝑨1=β‹―=𝑨𝑛=0

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SLIDE 109

Central Identity

Suppose 𝑀1, … , 𝑀𝑛 are independent random vectors with 𝐡𝑗 ≔ 𝔽𝑀𝑗𝑀𝑗

π‘ˆ. Then

𝔽det 𝑦𝐽 βˆ’

𝑗

𝑀𝑗𝑀𝑗

π‘ˆ

=

𝑗=1 𝑛

1 βˆ’ πœ– πœ–π‘¨π‘— det 𝑦𝐽 +

𝑗

𝑨𝑗𝐡𝑗

𝑨1=β‹―=𝑨𝑛=0

Proof: easy, tomorrow.

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SLIDE 110

Proof of Master Real- Rootedness Theorem

Suppose 𝑀1, … , 𝑀𝑛 are independent random vectors with 𝐡𝑗 ≔ 𝔽𝑀𝑗𝑀𝑗

π‘ˆ. Then

𝔽det 𝑦𝐽 βˆ’

𝑗

𝑀𝑗𝑀𝑗

π‘ˆ

=

𝑗=1 𝑛

1 βˆ’ πœ– πœ–π‘¨π‘— det 𝑦𝐽 +

𝑗

𝑨𝑗𝐡𝑗

𝑨1=β‹―=𝑨𝑛=0

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SLIDE 111

Proof of Master Real- Rootedness Theorem

Suppose 𝑀1, … , 𝑀𝑛 are independent random vectors with 𝐡𝑗 ≔ 𝔽𝑀𝑗𝑀𝑗

π‘ˆ. Then

𝔽det 𝑦𝐽 βˆ’

𝑗

𝑀𝑗𝑀𝑗

π‘ˆ

=

𝑗=1 𝑛

1 βˆ’ πœ– πœ–π‘¨π‘— det 𝑦𝐽 +

𝑗

𝑨𝑗𝐡𝑗

𝑨1=β‹―=𝑨𝑛=0

slide-112
SLIDE 112

Proof of Master Real- Rootedness Theorem

Suppose 𝑀1, … , 𝑀𝑛 are independent random vectors with 𝐡𝑗 ≔ 𝔽𝑀𝑗𝑀𝑗

π‘ˆ. Then

𝔽det 𝑦𝐽 βˆ’

𝑗

𝑀𝑗𝑀𝑗

π‘ˆ

=

𝑗=1 𝑛

1 βˆ’ πœ– πœ–π‘¨π‘— det 𝑦𝐽 +

𝑗

𝑨𝑗𝐡𝑗

𝑨1=β‹―=𝑨𝑛=0

slide-113
SLIDE 113

Proof of Master Real- Rootedness Theorem

Suppose 𝑀1, … , 𝑀𝑛 are independent random vectors with 𝐡𝑗 ≔ 𝔽𝑀𝑗𝑀𝑗

π‘ˆ. Then

𝔽det 𝑦𝐽 βˆ’

𝑗

𝑀𝑗𝑀𝑗

π‘ˆ

=

𝑗=1 𝑛

1 βˆ’ πœ– πœ–π‘¨π‘— det 𝑦𝐽 +

𝑗

𝑨𝑗𝐡𝑗

𝑨1=β‹―=𝑨𝑛=0

slide-114
SLIDE 114

Proof of Master Real- Rootedness Theorem

Suppose 𝑀1, … , 𝑀𝑛 are independent random vectors with 𝐡𝑗 ≔ 𝔽𝑀𝑗𝑀𝑗

π‘ˆ. Then

𝔽det 𝑦𝐽 βˆ’

𝑗

𝑀𝑗𝑀𝑗

π‘ˆ

=

𝑗=1 𝑛

1 βˆ’ πœ– πœ–π‘¨π‘— det 𝑦𝐽 +

𝑗

𝑨𝑗𝐡𝑗

𝑨1=β‹―=𝑨𝑛=0

slide-115
SLIDE 115

The Whole Proof

𝔽det 𝑦𝐽 βˆ’ 𝑗 𝑀𝑗𝑀𝑗

π‘ˆ is real-rooted for all indep. 𝑀𝑗.

slide-116
SLIDE 116

The Whole Proof

𝔽det 𝑦𝐽 βˆ’ 𝑗 𝑀𝑗𝑀𝑗

π‘ˆ is real-rooted for all indep. 𝑀𝑗.

π”½πœ“π΅π‘‘(𝑒 βˆ’ 𝑦) is real-rooted for all product distributions on signings.

slide-117
SLIDE 117

The Whole Proof

𝔽det 𝑦𝐽 βˆ’ 𝑗 𝑀𝑗𝑀𝑗

π‘ˆ is real-rooted for all indep. 𝑀𝑗.

π”½πœ“π΅π‘‘(𝑦) is real-rooted for all product distributions on signings.

slide-118
SLIDE 118

The Whole Proof

𝔽det 𝑦𝐽 βˆ’ 𝑗 𝑀𝑗𝑀𝑗

π‘ˆ is real-rooted for all indep. 𝑀𝑗.

π”½πœ“π΅π‘‘(𝑦) is real-rooted for all product distributions on signings. πœ“π΅π‘‘ 𝑦

π‘¦βˆˆ Β±1 𝑛 is an interlacing family

slide-119
SLIDE 119

The Whole Proof

𝔽det 𝑦𝐽 βˆ’ 𝑗 𝑀𝑗𝑀𝑗

π‘ˆ is real-rooted for all indep. 𝑀𝑗.

such that π”½πœ“π΅π‘‘(𝑦) is real-rooted for all product distributions on signings. πœ“π΅π‘‘ 𝑦

π‘¦βˆˆ Β±1 𝑛 is an interlacing family

slide-120
SLIDE 120

3-Step Proof Strategy

  • 1. Show that some poly does as well as the .
  • 2. Calculate the expected polynomial.
  • 3. Bound the largest root of the expected poly.

such that

οƒ– οƒ– οƒ–

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SLIDE 121

Infinite Sequences of Bipartite Ramanujan Graphs

Find an operation which doubles the size of a graph without blowing up its eigenvalues.

[ ]

  • d

d

[

  • 2 𝑒 βˆ’ 1

]

2 𝑒 βˆ’ 1

… ∞

slide-122
SLIDE 122

Main Theme

Reduced the existence of a good matrix to:

  • 1. Proving real-rootedness of an

expected polynomial.

  • 2. Bounding roots of the expected

polynomial.

slide-123
SLIDE 123

Main Theme

Reduced the existence of a good matrix to:

  • 1. Proving real-rootedness of an

expected polynomial. (general, using real stability)

  • 2. Bounding roots of the expected

polynomial. (specific, using combinatorics)

slide-124
SLIDE 124

Tomorrow

Reduced the existence of a good matrix to:

  • 1. Proving real-rootedness of an

expected polynomial. (general, using real stability)

  • 2. Bounding roots of the expected

polynomial. (general, using new method)

slide-125
SLIDE 125

Tomorrow

Reduced the existence of a good matrix to:

  • 1. Proving real-rootedness of an

expected polynomial. (general, using real stability)

  • 2. Bounding roots of the expected

polynomial. (general, using new method)

major implications in combinatorics, linear algebra + 1959 Kadison-Singer Conjecture.

slide-126
SLIDE 126

Tomorrow

Reduced the existence of a good matrix to:

  • 1. Proving real-rootedness of an

expected polynomial. (general, using real stability)

  • 2. Bounding roots of the expected

polynomial. (general, using new method)

major implications in combinatorics, linear algebra + 1959 Kadison-Singer Conjecture.