Proof that Expand using permutations x ± 1 0 0 ± 1 ± 1 same edge: ± 1 x ± 1 0 0 0 same value 0 ± 1 x ± 1 0 0 0 0 ± 1 x ± 1 0 ± 1 0 0 ± 1 x ± 1 ± 1 0 0 0 ± 1 x
Proof that Expand using permutations x ± 1 0 0 ± 1 ± 1 ± 1 x ± 1 0 0 0 0 ± 1 x ± 1 0 0 0 0 ± 1 x ± 1 0 ± 1 0 0 ± 1 x ± 1 ± 1 0 0 0 ± 1 x Get 0 if hit any 0s
Proof that Expand using permutations x ± 1 0 0 ± 1 ± 1 ± 1 x ± 1 0 0 0 0 ± 1 x ± 1 0 0 0 0 ± 1 x ± 1 0 ± 1 0 0 ± 1 x ± 1 ± 1 0 0 0 ± 1 x Get 0 if take just one entry for any edge
Proof that Expand using permutations x ± 1 0 0 ± 1 ± 1 ± 1 x ± 1 0 0 0 0 ± 1 x ± 1 0 0 0 0 ± 1 x ± 1 0 ± 1 0 0 ± 1 x ± 1 ± 1 0 0 0 ± 1 x Only permutations that count are involutions
Proof that Expand using permutations x ± 1 0 0 ± 1 ± 1 ± 1 x ± 1 0 0 0 0 ± 1 x ± 1 0 0 0 0 ± 1 x ± 1 0 ± 1 0 0 ± 1 x ± 1 ± 1 0 0 0 ± 1 x Only permutations that count are involutions
Proof that Expand using permutations x ± 1 0 0 ± 1 ± 1 ± 1 x ± 1 0 0 0 0 ± 1 x ± 1 0 0 0 0 ± 1 x ± 1 0 ± 1 0 0 ± 1 x ± 1 ± 1 0 0 0 ± 1 x Only permutations that count are involutions Correspond to matchings
Proof that Expand using permutations x ± 1 0 0 ± 1 ± 1 ± 1 x ± 1 0 0 0 0 ± 1 x ± 1 0 0 0 0 ± 1 x ± 1 0 ± 1 0 0 ± 1 x ± 1 ± 1 0 0 0 ± 1 x Only permutations that count are involutions Correspond to matchings
3-Step Proof Strategy 1. Show that some poly does as well as the . such that 2. Calculate the expected polynomial. [Godsil- Gutman’81] 3. Bound the largest root of the expected poly.
3-Step Proof Strategy 1. Show that some poly does as well as the . such that 2. Calculate the expected polynomial. [Godsil- Gutman’81] 3. Bound the largest root of the expected poly.
The matching polynomial (Heilmann-Lieb ‘72) Theorem (Heilmann-Lieb) all the roots are real
The matching polynomial (Heilmann-Lieb ‘72) Theorem (Heilmann-Lieb) all the roots are real and have absolute value at most Proof : simple, based on recurrences.
3-Step Proof Strategy 1. Show that some poly does as well as the . such that 2. Calculate the expected polynomial. [Godsil- Gutman’81] 3. Bound the largest root of the expected poly. [Heilmann- Lieb’72]
3-Step Proof Strategy 1. Show that some poly does as well as the . such that 2. Calculate the expected polynomial. [Godsil- Gutman’81] 3. Bound the largest root of the expected poly. [Heilmann- Lieb’72]
3-Step Proof Strategy 1. Show that some poly does as well as the . such that Implied by: “ is an interlacing family.”
Averaging Polynomials Basic Question : Given when are the roots of the related to roots of ?
Averaging Polynomials Basic Question : Given when are the roots of the related to roots of ? Answer: Certainly not always
Averaging Polynomials Basic Question : Given when are the roots of the related to roots of ? Answer: Certainly not always…
Averaging Polynomials Basic Question : Given when are the roots of the related to roots of ? But sometimes it works:
A Sufficient Condition Basic Question : Given when are the roots of the related to roots of ? Answer: When they have a common interlacing . Definition. interlaces if
Theorem. If have a common interlacing,
Theorem. If have a common interlacing, Proof.
Theorem. If have a common interlacing, Proof.
Theorem. If have a common interlacing, Proof.
Theorem. If have a common interlacing, Proof.
Theorem. If have a common interlacing, Proof.
Theorem. If have a common interlacing, Proof.
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
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
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
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
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
Interlacing Family of Polynomials Theorem: There is an s so that
Interlacing Family of Polynomials Theorem: There is an s so that Proof: By common interlacing, one of , has
Interlacing Family of Polynomials Theorem: There is an s so that Proof: By common interlacing, one of , has
Interlacing Family of Polynomials Theorem: There is an s so that Proof: By common interlacing, one of , has
Interlacing Family of Polynomials Theorem: There is an s so that Proof: By common interlacing, one of , has ….
Interlacing Family of Polynomials Theorem: There is an s so that Proof: By common interlacing, one of , has
An interlacing family Theorem : Let is an interlacing family
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
To prove interlacing family Let Leaves of tree = signings 𝑡 1 , … , 𝑡 𝑛 Internal nodes = partial signings 𝑡 1 , … , 𝑡 𝑙
To prove interlacing family Let Need to prove that for all , is real rooted
To prove interlacing family Let Need to prove that for all , is real rooted are fixed is 1 with probability , -1 with are uniformly
Generalization of Heilmann-Lieb Suffices to prove that is real rooted for every independent distribution on the entries of s
Generalization of Heilmann-Lieb Suffices to prove that is real rooted for every independent distribution on the entries of s:
Transformation to PSD Matrices Suffices to show real rootedness of
Transformation to PSD Matrices Suffices to show real rootedness of Why is this useful?
Transformation to PSD Matrices Suffices to show real rootedness of Why is this useful?
Transformation to PSD Matrices
Transformation to PSD Matrices 𝑈 𝔽 𝑡 det 𝑦𝐽 − 𝑒𝐽 − 𝐵 𝑡 = 𝔽det 𝑦𝐽 − 𝑤 𝑗𝑘 𝑤 𝑗𝑘 𝑗𝑘∈𝐹 𝜀 𝑗 − 𝜀 𝑘 with probability λ 𝑗𝑘 where 𝑤 𝑗𝑘 = 𝜀 𝑗 + 𝜀 𝑘 with probability (1−𝜇 𝑗𝑘 )
Master Real-Rootedness Theorem Given any independent random vectors 𝑤 1 , … , 𝑤 𝑛 ∈ ℝ 𝑒 , their expected characteristic polymomial 𝑈 𝔽det 𝑦𝐽 − 𝑤 𝑗 𝑤 𝑗 𝑗 has real roots.
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