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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results Bounding entropies of hard squares and friends How to pick a good vector Andrew Rechnitzer Yao-ban Chan Melbourne, April 2013 Rechnitzer Packing bits 2d Bounds Upper

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

Bounding entropies of hard squares and friends

How to pick a good vector Andrew Rechnitzer Yao-ban Chan Melbourne, April 2013

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

INE YE OLDE DÆS

In the dark ages there was tape.

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

INE YE OLDE DÆS

In the dark ages there was tape. Data is stored along tape as magnetised regions.

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

FIELD UP, FIELD DOWN, ONE AND ZERO

Naive idea — store 1’s and 0’s as regions with field in different directions.

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

FIELD UP, FIELD DOWN, ONE AND ZERO

Naive idea — store 1’s and 0’s as regions with field in different directions.

A core question

How much data can we store?

Rechnitzer

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SLIDE 6

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

FIELD UP, FIELD DOWN, ONE AND ZERO

Naive idea — store 1’s and 0’s as regions with field in different directions.

A core question

How much data can we store?

  • n regions can store 2n possible words.
  • 1 bit per region.

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

REAL WORLD GETS IN THE WAY

The engineering is easier if we encode data as

  • Store 0 as “field unchanged”
  • Store 1 as “field changed”

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

FLIP-FLOP PROBLEMS

  • The magnetic regions are not perfectly discrete
  • The read mechanism might misread “change-change”.

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

ENCODE DATA DIFFERENTLY

  • Store data so that we forbid “change-change”
  • Store words in {0, 1} so that there is no “11” subword.

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

ENCODE DATA DIFFERENTLY

  • Store data so that we forbid “change-change”
  • Store words in {0, 1} so that there is no “11” subword.

A core question

How much data can we store? How many legal words are there?

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

COUNT LEGAL WORDS

Let

  • ψn(⊕) be # legal words ending in ⊕
  • ψn(⊖) be # legal words ending in ⊖

ψn+1(⊕) = ψn(⊖) ψn+1(⊖) = ψn(⊕) + ψn(⊖)

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

COUNT LEGAL WORDS

Let

  • ψn(⊕) be # legal words ending in ⊕
  • ψn(⊖) be # legal words ending in ⊖

ψn+1(⊕) = ψn(⊖) ψn+1(⊖) = ψn(⊕) + ψn(⊖) = ψn

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

COUNT LEGAL WORDS

Let

  • ψn(⊕) be # legal words ending in ⊕
  • ψn(⊖) be # legal words ending in ⊖

ψn+1(⊕) = ψn(⊖) ψn+1(⊖) = ψn(⊕) + ψn(⊖) = ψn ψn+1(⊖) = ψn(⊖) + ψn−1(⊖) ψn = ψn−1 + ψn−2

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

BUILD A TRANSFER MATRIX

More generally. . . ψn+1(⊖) ψn+1(⊕)

  • =

1 1 1 ψn(⊖) ψn(⊕)

  • Rechnitzer
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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

BUILD A TRANSFER MATRIX

More generally. . . ψn+1(⊖) ψn+1(⊕)

  • =

1 1 1 ψn(⊖) ψn(⊕)

  • =

1 1 1 n ψ0(⊖) ψ0(⊕)

  • Rechnitzer
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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

BUILD A TRANSFER MATRIX

More generally. . . ψn+1(⊖) ψn+1(⊕)

  • =

1 1 1 ψn(⊖) ψn(⊕)

  • =

1 1 1 n ψ0(⊖) ψ0(⊕)

  • = PT

λn

1

λn

2

  • P

1 1

  • Number of words ∼ nth power of dominant eigenvalue

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

1D IS EASY

So for this “11”-forbidden model ψn ∼ 1 + √ 5 2 n Entropy of encoding is log2

  • 1+

√ 5 2

  • ≈ 0.69 bits per region.

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

1D IS EASY

So for this “11”-forbidden model ψn ∼ 1 + √ 5 2 n Entropy of encoding is log2

  • 1+

√ 5 2

  • ≈ 0.69 bits per region.

What about other models?

  • Run-length limited (d, k)

— forbid subwords {11, 101, 1001, . . . 10d1, 0k+1}.

  • Charge model (b)

— cumulative charge lies between ±b.

  • Parity models

— even # 0’s between 1’s. — odd # 0’s between 1’s. Use same transfer matrix machinery.

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

BUT NOW WE LIVE IN THE FUTURE. . .

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

BUT NOW WE LIVE IN THE FUTURE. . .

and we can store data in 2d! (InPhase Technologies & hVault)

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

BUT NOW WE LIVE IN THE FUTURE. . .

and we can store data in 2d! (InPhase Technologies & hVault) Coding theorists extend entropy question from 1d to 2d

A core question

How much data can we store in 2d?

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

BUT NOW WE LIVE IN THE FUTURE. . .

and we can store data in 2d! (InPhase Technologies & hVault) Coding theorists extend entropy question from 1d to 2d

A core question

How many 2d words avoid 11 and 1 1 ?

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

WHAT DOES THIS LOOK LIKE?

Rechnitzer

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SLIDE 24

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

WHAT DOES THIS LOOK LIKE?

2d coding problem = hard square lattice gas

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

WHAT DOES THIS LOOK LIKE?

2d coding problem = hard square lattice gas = independent sets on Z2

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

WHAT DO WE WANT TO KNOW?

More generally. . .

2d shift of finite type

  • Given a finite alphabet A, and
  • a finite set of words F,
  • a word in AZ2 is valid when it avoids words in F.

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

WHAT DO WE WANT TO KNOW?

More generally. . .

2d shift of finite type

  • Given a finite alphabet A, and
  • a finite set of words F,
  • a word in AZ2 is valid when it avoids words in F.

Entropy

  • Let Cn×n be the # valid n × n words.
  • Entropy is log2 κ = lim

n→∞

1 n2 log2 Cn×n

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

WHAT DO WE WANT TO KNOW?

More generally. . .

2d shift of finite type

  • Given a finite alphabet A, and
  • a finite set of words F,
  • a word in AZ2 is valid when it avoids words in F.

Entropy

  • Let Cn×n be the # valid n × n words.
  • Entropy is log2 κ = lim

n→∞

1 n2 log2 Cn×n So what do we know. . .

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

PROVABLY HARD

  • Algorithmically undecideable if there are any valid words

[Berger 1966]

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

PROVABLY HARD

  • Algorithmically undecideable if there are any valid words

[Berger 1966]

  • In 1d, κ ∈ R+ is an entropy iff κ is a Peron number

[Lind 1983]

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

PROVABLY HARD

  • Algorithmically undecideable if there are any valid words

[Berger 1966]

  • In 1d, κ ∈ R+ is an entropy iff κ is a Peron number

[Lind 1983]

  • In 2d and up, κ ∈ R+ is an entropy iff κ is recursively enumerable

[Hochman & Meyerovitch 2007]

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

PROVABLY HARD

  • Algorithmically undecideable if there are any valid words

[Berger 1966]

  • In 1d, κ ∈ R+ is an entropy iff κ is a Peron number

[Lind 1983]

  • In 2d and up, κ ∈ R+ is an entropy iff κ is recursively enumerable

[Hochman & Meyerovitch 2007]

  • In 2d and up, κ known exactly for very few SFTs

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

EXAMPLE OF EXACT

Odd constraint

Words in {0, 1} so that between 1’s there are odd number of 0’s. [Louidor & Marcus 2010] κ = √ 2.

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

EXAMPLE OF EXACT

Odd constraint

Words in {0, 1} so that between 1’s there are odd number of 0’s. [Louidor & Marcus 2010] κ = √ 2. One sub-lattice fixed as 0’s and other is unconstrained.

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

BOUNDS

Back to hardsquares.. .

  • No reason that κ should have a “nice” expression.
  • So try to find tight bounds.

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

BOUNDS

Back to hardsquares.. .

  • No reason that κ should have a “nice” expression.
  • So try to find tight bounds.

Most approaches based on transfer matrices Big problem — # states grows exponentially with width

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

TRANSFER MATRIX

Tw = column-to-column TM for hard squares in strip of width w

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

TRANSFER MATRIX

Tw = column-to-column TM for hard squares in strip of width w

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

TRANSFER MATRIX

Tw = column-to-column TM for hard squares in strip of width w κ = lim

w→∞ Λ1/w w

where Λw is dominant eigenvalue

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

USEFUL IDEAS FROM LINEAR ALGEBRA 101

Symmetric matrix V

  • Eigenvalues λ1, . . . , λn all real

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

USEFUL IDEAS FROM LINEAR ALGEBRA 101

Symmetric matrix V

  • Eigenvalues λ1, . . . , λn all real
  • Min-max Theorem — for any non-trivial vector x,

λmin ≤ x | V | x x | x ≤ λmax

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

USEFUL IDEAS FROM LINEAR ALGEBRA 101

Symmetric matrix V

  • Eigenvalues λ1, . . . , λn all real
  • Min-max Theorem — for any non-trivial vector x,

λmin ≤ x | V | x x | x ≤ λmax

  • Trace of power

Tr Vk = λk

1 + λk 2 + · · · + λk n

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

USEFUL IDEAS FROM LINEAR ALGEBRA 101

Symmetric matrix V

  • Eigenvalues λ1, . . . , λn all real
  • Min-max Theorem — for any non-trivial vector x,

λmin ≤ x | V | x x | x ≤ λmax

  • Trace of power

Tr Vk = λk

1 + λk 2 + · · · + λk n

Tr V2k = λ2k

1 + λ2k 2 + · · · + λ2k n ≥ λ2k max

Rechnitzer

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SLIDE 44

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

USEFUL IDEAS FROM LINEAR ALGEBRA 101

Symmetric matrix V

  • Eigenvalues λ1, . . . , λn all real
  • Min-max Theorem — for any non-trivial vector x,

λmin ≤ x | V | x x | x ≤ λmax

  • Trace of power

Tr Vk = λk

1 + λk 2 + · · · + λk n

Tr V2k = λ2k

1 + λ2k 2 + · · · + λ2k n ≥ λ2k max

Leverage these to get good bounds [Engel 1990] and [Calkin & Wilf 1998]

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

TRACE TRICK

Rewrite trace Tr V2k =

  • Vψ0,ψ1Vψ1,ψ2 . . . Vψ2k−1,ψ0

Sum is over all sequences of states, but only “legal” ones count

Rechnitzer

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SLIDE 46

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

TRACE TRICK

Rewrite trace Tr V2k =

  • Vψ0,ψ1Vψ1,ψ2 . . . Vψ2k−1,ψ0

Sum is over all sequences of states, but only “legal” ones count

Rechnitzer

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SLIDE 47

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

TRACE TRICK

Rewrite trace Tr V2k =

  • Vψ0,ψ1Vψ1,ψ2 . . . Vψ2k−1,ψ0

Sum is over all sequences of states, but only “legal” ones count So Tr T2k

w is equivalent to “legal” configurations on rings

Tr T2k

w =

  • 1
  • Bw−1

2k

  • 1
  • Rechnitzer
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SLIDE 48

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

TRACE TRICK

Rewrite trace Tr V2k =

  • Vψ0,ψ1Vψ1,ψ2 . . . Vψ2k−1,ψ0

Sum is over all sequences of states, but only “legal” ones count So Tr T2k

w is equivalent to “legal” configurations on rings

Tr T2k

w =

  • 1
  • Bw−1

2k

  • 1
  • Sneaky — “width” is now exponent.

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

LIMITS

So build TM for rings B2k — also grows exponentially with circumference.

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

LIMITS

So build TM for rings B2k — also grows exponentially with circumference. Λ2k

w ≤ Tr T2k w =

  • 1
  • Bw−1

2k

  • 1
  • Rechnitzer
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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

LIMITS

So build TM for rings B2k — also grows exponentially with circumference. Λ2k

w ≤ Tr T2k w =

  • 1
  • Bw−1

2k

  • 1
  • Raise to 1/w and let width → ∞

Λ2k/w

w

≤

  • Tr T2k

w

1/w =

  • 1
  • Bw−1

2k

  • 1

1/w ↓ ↓ κ2k ≤ ξ2k

Upper bound

Let B2k be the TM for system on ring of circumference 2k, then κ ≤ ξ1/2k

2k

where ξ2k is dominant eigenvalue of B2k.

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

RESULTS

ξ2 = 2.41421356237309504 . . . κ ≤ 1.55377397403003730 . . .

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

RESULTS

ξ2 = 2.41421356237309504 . . . κ ≤ 1.55377397403003730 . . . ξ4 = 5.15632517465866169 . . . κ ≤ 1.50690222590181180 . . .

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

RESULTS

ξ2 = 2.41421356237309504 . . . κ ≤ 1.55377397403003730 . . . ξ4 = 5.15632517465866169 . . . κ ≤ 1.50690222590181180 . . . ξ6 = 11.5517095660481450 . . . κ ≤ 1.50351480947590302 . . . [Calkin & Wilf 1998]

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

RESULTS

ξ2 = 2.41421356237309504 . . . κ ≤ 1.55377397403003730 . . . ξ4 = 5.15632517465866169 . . . κ ≤ 1.50690222590181180 . . . ξ6 = 11.5517095660481450 . . . κ ≤ 1.50351480947590302 . . . [Calkin & Wilf 1998] ξ36 = 2349759.74655388695 . . . κ ≤ 1.5030480824753399273 [Friedland, Lundow & Markstr¨

  • m 2010]

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

RESULTS

ξ2 = 2.41421356237309504 . . . κ ≤ 1.55377397403003730 . . . ξ4 = 5.15632517465866169 . . . κ ≤ 1.50690222590181180 . . . ξ6 = 11.5517095660481450 . . . κ ≤ 1.50351480947590302 . . . [Calkin & Wilf 1998] ξ36 = 2349759.74655388695 . . . κ ≤ 1.5030480824753399273 [Friedland, Lundow & Markstr¨

  • m 2010]

Huge transfer matrix — use symmetries to compress it.

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

RAYLEIGH QUOTIENTS

Min-max theorem

λmin ≤ x | V | x x | x ≤ λmax

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

RAYLEIGH QUOTIENTS

Min-max theorem

λmin ≤ x | V | x x | x ≤ λmax So the simplest idea — set |x = |1.

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

RAYLEIGH QUOTIENTS

Min-max theorem

λmin ≤ x | V | x x | x ≤ λmax So the simplest idea — set |x = |1. Λw ≥ 1 | Tw | 1 1 | 1

Rechnitzer

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SLIDE 60

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

RAYLEIGH QUOTIENTS

Min-max theorem

λmin ≤ x | V | x x | x ≤ λmax So the simplest idea — set |x = |1. Λw ≥ 1 | Tw | 1 1 | 1 For fixed w this is silly — instead compute the eigenvalue by power method.

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

RAYLEIGH QUOTIENTS

Min-max theorem

λmin ≤ x | V | x x | x ≤ λmax So the simplest idea — set |x = |1. Λw ≥ 1 | Tw | 1 1 | 1 For fixed w this is silly — instead compute the eigenvalue by power method. But if we can choose a better vector. . .

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

SNEAKY TRICKS AGAIN

Vector |1 a poor choice. Λp

w ≥

  • 1
  • Tp

w

  • 1
  • 1 | 1

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

SNEAKY TRICKS AGAIN

Power method — replace |1 with Tq

w |1.

Λp

w ≥

  • Tq

w1

  • Tp

w

  • Tq

w1

  • Tq

w1

  • Tq

w1

  • Rechnitzer
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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

SNEAKY TRICKS AGAIN

Massage denominator

  • Tq

w1

  • Tq

w1

  • =
  • 1
  • T2q

w

  • 1
  • Rechnitzer
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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

SNEAKY TRICKS AGAIN

Massage denominator

  • Tq

w1

  • Tq

w1

  • =
  • 1
  • T2q

w

  • 1
  • Rechnitzer
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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

SNEAKY TRICKS AGAIN

All configs in w × 2q rectangle = configs in 2q × w rectangle

  • Tq

w1

  • Tq

w1

  • =
  • 1
  • T2q

w

  • 1
  • =
  • 1
  • Tw

2q

  • 1
  • Sneaky — width becomes exponent

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

SNEAKY TRICKS AGAIN

Look at numerator now Λp

w ≥

  • Tq

w1

  • Tp

w

  • Tq

w1

  • Tq

w1

  • Tq

w1

  • Rechnitzer
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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

SNEAKY TRICKS AGAIN

Massage things a little

  • Tq

w1

  • Tp

w

  • Tq

w1

  • =
  • 1
  • T2q+p

w

  • 1
  • Rechnitzer
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SLIDE 69

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

SNEAKY TRICKS AGAIN

Massage things a little

  • Tq

w1

  • Tp

w

  • Tq

w1

  • =
  • 1
  • T2q+p

w

  • 1
  • Rechnitzer
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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

SNEAKY TRICKS AGAIN

Again use the x ↔ y symmetry

  • Tq

w1

  • Tp

w

  • Tq

w1

  • =
  • 1
  • T2q+p

w

  • 1
  • =
  • 1
  • Tw

2q+p

  • 1
  • Sneaky — width becomes exponent

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

RESULTS

Putting this together Λp

w ≥

  • 1
  • Tw

2q+p

  • 1
  • 1
  • Tw

2q

  • 1
  • Now raise to 1/w and let w → ∞

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

RESULTS

Putting this together Λp

w ≥

  • 1
  • Tw

2q+p

  • 1
  • 1
  • Tw

2q

  • 1
  • Now raise to 1/w and let w → ∞

Lower bound [Calkin & Wilf 1998]

For any p, q ≥ 1 κp ≥ Λ2q+p Λ2q

Rechnitzer

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

RESULTS

Putting this together Λp

w ≥

  • 1
  • Tw

2q+p

  • 1
  • 1
  • Tw

2q

  • 1
  • Now raise to 1/w and let w → ∞

Lower bound [Calkin & Wilf 1998]

For any p, q ≥ 1 κp ≥ Λ2q+p Λ2q

  • κ ≥ 1.50304768131466259 . . . (p = 3, q = 2)

[Calkin & Wilf]

  • κ ≥ 1.50304808247533226 . . . (p = 1, q = 13)

[Friedland et al]

Rechnitzer

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SLIDE 74

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

PICK A BETTER VECTOR

  • We use corner transfer matrix formalism to pick a better vector.

Rechnitzer

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SLIDE 75

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

PICK A BETTER VECTOR

  • We use corner transfer matrix formalism to pick a better vector.
  • Corner transfer matrices used to study lattice gas & magnet models

[Baxter 1968]

Rechnitzer

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SLIDE 76

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

PICK A BETTER VECTOR

  • We use corner transfer matrix formalism to pick a better vector.
  • Corner transfer matrices used to study lattice gas & magnet models

[Baxter 1968]

  • Very famously lead to solution of hard hexagons [Baxter 1980]

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SLIDE 77

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

HOW TO BUILD A VECTOR

Each entry of vector corresponds to a state along the cut

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SLIDE 78

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

HOW TO BUILD A VECTOR

Each entry of vector corresponds to a state along the cut

Baxter’s Ansatz which extends [Kramers & Wannier 1941]

Build Rayleigh quotient with vector ψ ψ(σ1, σ2, . . . , σw) = Tr [F(σ1, σ2)F(σ2, σ3) . . . F(σw, σ1)] For some matrices F(a, b).

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SLIDE 79

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

WHAT DOES THIS LOOK LIKE?

  • Can think of F as a “literal” half-row transfer matrix.

— but it can be almost any matrix.

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SLIDE 80

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

WHAT DOES THIS LOOK LIKE?

  • Can think of F as a “literal” half-row transfer matrix.

— but it can be almost any matrix.

  • Trace makes it a cylinder — doesn’t change bound.

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

RAYLEIGH QUOTIENT → TRACES

Rayleigh quotient

Λw ≥ ψ | Tw | ψ ψ | ψ ψ | T | ψ = Tr Sw ψ | ψ = Tr Rw

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SLIDE 82

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

RAYLEIGH QUOTIENT → TRACES

Rayleigh quotient

Λw ≥ ψ | Tw | ψ ψ | ψ ψ | T | ψ = Tr Sw ψ | ψ = Tr Rw

Rechnitzer

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SLIDE 83

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

RAYLEIGH QUOTIENT → TRACES

Rayleigh quotient

Λw ≥ ψ | Tw | ψ ψ | ψ ψ | T | ψ = Tr Sw ψ | ψ = Tr Rw Where ω = 1 if face valid else ω = 0.

Rechnitzer

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SLIDE 84

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

TO GET A BOUND

Lower bound

κ = lim

w→∞ Λ1/w w

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SLIDE 85

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

TO GET A BOUND

Lower bound

κ = lim

w→∞ Λ1/w w

≥ lim

w→∞

Tr Sw Tr Rw 1/w

Rechnitzer

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SLIDE 86

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

TO GET A BOUND

Lower bound

κ = lim

w→∞ Λ1/w w

≥ lim

w→∞

Tr Sw Tr Rw 1/w = η ξ where ξ, η are dominant eigenvalues of R and S.

Rechnitzer

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SLIDE 87

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

TO GET A BOUND

Lower bound

κ = lim

w→∞ Λ1/w w

≥ lim

w→∞

Tr Sw Tr Rw 1/w = η ξ where ξ, η are dominant eigenvalues of R and S.

1 Pick matrices F — note dimension need not be related to w 2 Form matrices R and S 3 Compute dominant eigenvalues of ξ, η.

Rechnitzer

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SLIDE 88

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

TO GET A BOUND

Lower bound

κ = lim

w→∞ Λ1/w w

≥ lim

w→∞

Tr Sw Tr Rw 1/w = η ξ where ξ, η are dominant eigenvalues of R and S.

1 Pick matrices F — note dimension need not be related to w 2 Form matrices R and S 3 Compute dominant eigenvalues of ξ, η.

But how do we pick F?

Rechnitzer

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SLIDE 89

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

TO GET A BOUND

Lower bound

κ = lim

w→∞ Λ1/w w

≥ lim

w→∞

Tr Sw Tr Rw 1/w = η ξ where ξ, η are dominant eigenvalues of R and S.

1 Pick matrices F — note dimension need not be related to w 2 Form matrices R and S 3 Compute dominant eigenvalues of ξ, η.

But how do we pick F? And where are these infamous “corner transfer matrices”?

Rechnitzer

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SLIDE 90

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

EIGENVECTORS → EIGENMATRICES(?)

R |X = ξ |X S |Y = η |Y |X , |Y eigenvectors of R and S.

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SLIDE 91

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

EIGENVECTORS → EIGENMATRICES(?)

R |X = ξ |X

  • b

F(a, b)X(b)F(b, a) = ξX(a) S |Y = η |Y

  • c,d

ω a b c d

  • F(a, c)Y(c, d)F(d, b) = ηY(a, b)

X(a), Y(a, b) ≈ “half-plane transfer matrices”

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SLIDE 92

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

TO MAXIMISE, PLANES → CORNER×CORNER

Baxter showed that Rayleigh quotient stationary when X(a) = A(a)2 Y(a, b) = A(a)F(a, b)A(b) where A is half of X — a “corner transfer matrix” Baxter then carefully picked F to make things work.

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SLIDE 93

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

RENORMALISE INSTEAD

  • We have used “corner transfer matrix renormalisation group method”

[Nishino & Okunishi 1996]

  • Related to density matrix renormalisation group method

[White 1992]

Rechnitzer

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SLIDE 94

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

RENORMALISE INSTEAD

  • We have used “corner transfer matrix renormalisation group method”

[Nishino & Okunishi 1996]

  • Related to density matrix renormalisation group method

[White 1992]

  • The central idea = only keep important parts of A.

Rechnitzer

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SLIDE 95

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

BUILD RECURSIVELY

Start by building “literal” matrices. Let

  • A be corner transfer matrix for a 2 × 2 grid
  • F be the half-row / half-column transfer matrix for a 1 × 2 grid

Rechnitzer

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SLIDE 96

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

BUILD RECURSIVELY

Start by building “literal” matrices. Let

  • A be corner transfer matrix for a 2 × 2 grid
  • F be the half-row / half-column transfer matrix for a 1 × 2 grid
  • Then build larger matrices by

Al(c)|d,a =

  • d

ω a b c d

  • F(c, d)A(b)F(b, a)

F(c, d)|b,a = ω a b c d

  • F(b, a)

Rechnitzer

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SLIDE 97

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

BUILD RECURSIVELY

Start by building “literal” matrices. Let

  • A be corner transfer matrix for a 2 × 2 grid
  • F be the half-row / half-column transfer matrix for a 1 × 2 grid
  • Then build larger matrices by

Al(c)|d,a =

  • d

ω a b c d

  • F(c, d)A(b)F(b, a)

F(c, d)|b,a = ω a b c d

  • F(b, a)
  • Iterate until A and F are huge — they are still “literal”.

Rechnitzer

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SLIDE 98

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

NOW ESTIMATE EIGENVALUES ξ, η

  • Look at eigenvalue equation:

ξ

  • a

X(a) =

  • a,b

F(a, b)X(b)F(b, a)

Rechnitzer

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SLIDE 99

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

NOW ESTIMATE EIGENVALUES ξ, η

  • Look at eigenvalue equation:

ξ

  • a

A(a)2 =

  • a,b

F(a, b)A(b)2F(b, a)

Rechnitzer

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SLIDE 100

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

NOW ESTIMATE EIGENVALUES ξ, η

  • Look at eigenvalue equation:

ξ

  • a

A(a)4 =

  • a,b

A(a)F(a, b)A(b)2F(b, a)A(a)

Rechnitzer

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SLIDE 101

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

NOW ESTIMATE EIGENVALUES ξ, η

  • Look at eigenvalue equation:

ξ Tr

  • a

A(a)4 = Tr

  • a,b

A(a)F(a, b)A(b)2F(b, a)A(a)

Rechnitzer

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SLIDE 102

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

NOW ESTIMATE EIGENVALUES ξ, η

  • Look at eigenvalue equation:

ξ = Tr

a,b A(a)F(a, b)A(b)2F(b, a)A(a)

Tr

a A(a)4

  • Invariant under similarity transform, so can diagonalise A.

Rechnitzer

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SLIDE 103

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

NOW ESTIMATE EIGENVALUES ξ, η

  • Look at eigenvalue equation:

ξ = Tr

a,b A(a)F(a, b)A(b)2F(b, a)A(a)

Tr

a A(a)4

  • Invariant under similarity transform, so can diagonalise A.
  • Key idea: discard small eigenvalues

Huge “literal” A, F → small “aphysical” A, F.

Rechnitzer

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SLIDE 104

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

NOW ESTIMATE EIGENVALUES ξ, η

  • Look at eigenvalue equation:

ξ = Tr

a,b A(a)F(a, b)A(b)2F(b, a)A(a)

Tr

a A(a)4

  • Invariant under similarity transform, so can diagonalise A.
  • Key idea: discard small eigenvalues

Huge “literal” A, F → small “aphysical” A, F.

Clever idea [Nishino & Okunishi 1996]

  • Building huge literal A, F and then projecting it down is wasteful.
  • Instead grow & project frequently until A, F converge.

Rechnitzer

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SLIDE 105

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

PUT IT ALL TOGETHER

1 Start with “reasonable” A, F. 2 Grow & project repeatedly until A, F converge. 3 Use this F to compute ξ, η and so lower bound for κ. 4 Grow A, F a little larger and repeat from #2.

Rechnitzer

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SLIDE 106

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

PUT IT ALL TOGETHER

1 Start with “reasonable” A, F. 2 Grow & project repeatedly until A, F converge. 3 Use this F to compute ξ, η and so lower bound for κ. 4 Grow A, F a little larger and repeat from #2.

Lower bound [YBC & AR]

κ ≥ 1. 5030480824753322643220 Previous best lower bound [Friedland, Lundow & Markstr¨

  • m 2010]

Rechnitzer

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SLIDE 107

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

PUT IT ALL TOGETHER

1 Start with “reasonable” A, F. 2 Grow & project repeatedly until A, F converge. 3 Use this F to compute ξ, η and so lower bound for κ. 4 Grow A, F a little larger and repeat from #2.

Lower bound [YBC & AR]

κ ≥ 1. 503048082475332264322066329475 55368938578103861030506202810 Previous best lower bound [Friedland, Lundow & Markstr¨

  • m 2010]

Previous best estimate [Baxter 1999]

Rechnitzer

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SLIDE 108

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

PUT IT ALL TOGETHER

1 Start with “reasonable” A, F. 2 Grow & project repeatedly until A, F converge. 3 Use this F to compute ξ, η and so lower bound for κ. 4 Grow A, F a little larger and repeat from #2.

Lower bound [YBC & AR]

κ ≥ 1. 503048082475332264322066329475 553689385781038610305062028101 73593385039692344038046329947 Previous best lower bound [Friedland, Lundow & Markstr¨

  • m 2010]

Previous best estimate [Baxter 1999] Our lower bound

Rechnitzer

slide-109
SLIDE 109

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

PUT IT ALL TOGETHER

1 Start with “reasonable” A, F. 2 Grow & project repeatedly until A, F converge. 3 Use this F to compute ξ, η and so lower bound for κ. 4 Grow A, F a little larger and repeat from #2.

Lower bound [YBC & AR]

κ ≥ 1. 503048082475332264322066329475 553689385781038610305062028101 73593385039692344038046329965 Previous best lower bound [Friedland, Lundow & Markstr¨

  • m 2010]

Previous best estimate [Baxter 1999] Our lower bound Our best estimate same except last 2 digits.

Rechnitzer

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SLIDE 110

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

OTHER MODELS

Hard squares, Read-write Isolated Memory and Non-Attacking Kings

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SLIDE 111

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

OTHER MODELS

Even model

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SLIDE 112

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

OTHER MODELS

Charge 3

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SLIDE 113

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

OTHER MODELS

Charge 3

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SLIDE 114

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

RESULTS

Substantial improvement of all previous lower bounds Model Matrix size Lower bound on (and estimate of) κ NAK 256 1.342 643 951 124 601 297 851 730 161 875 740 395 719 438 196 938 393 943 434 885 455 0 (1) RWIM 128 1.448 957 371 775 608 489 872 231 406 108 136 686 434 371 (7) Even 128 1.357 587 502 184 123 (5) Charge(3) 74 1.357 587 50 4-Colouring 96 2.336 056 641 041 133 656 814 01 (4) 5-Colouring 64 3.250 404 923 167 119 143 819 73 (6)

  • NAK, RWIM, Even, Charge(3) — [Louidor & Marcus (2010)]
  • 4-Colouring and 5-colouring — [Lundow & Markstr¨
  • m (2008)]

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Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

RESULTS

Substantial improvement of all previous lower bounds Model Matrix size Lower bound on (and estimate of) κ NAK 256 1.342 643 951 124 601 297 851 730 161 875 740 395 719 438 196 938 393 943 434 885 455 0 (1) RWIM 128 1.448 957 371 775 608 489 872 231 406 108 136 686 434 371 (7) Even 128 1.357 587 502 184 123 (5) Charge(3) 74 1.357 587 50 4-Colouring 96 2.336 056 641 041 133 656 814 01 (4) 5-Colouring 64 3.250 404 923 167 119 143 819 73 (6)

  • NAK, RWIM, Even, Charge(3) — [Louidor & Marcus (2010)]
  • 4-Colouring and 5-colouring — [Lundow & Markstr¨
  • m (2008)]

Why are Even and Charge(3) the same?

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SLIDE 116

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

OPEN QUESTIONS

  • What other models?

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SLIDE 117

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

OPEN QUESTIONS

  • What other models?
  • Upper bounds?

Rechnitzer

slide-118
SLIDE 118

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

OPEN QUESTIONS

  • What other models?
  • Upper bounds?
  • Methods in literature require computing eigenvalues of huge matrices

Can we find a method that relies on picking a good vector?

Rechnitzer

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SLIDE 119

Packing bits 2d Bounds Upper Lower Picking well Beware CTM Results

OPEN QUESTIONS

  • What other models?
  • Upper bounds?
  • Methods in literature require computing eigenvalues of huge matrices

Can we find a method that relies on picking a good vector?

Bounds due to [Collatz 1942]

If T is non-negative and x is any positive vector, then min

i

  • (Tx)i

xi

  • ≤ Λ ≤ max

i

  • (Tx)i

xi

  • Rechnitzer