text: 8 Unfortunately, even with this optimal choice of marker length, using deduplication chunks that have lengths of different order than the source blocks can lead to asymptotically signiﬁcantly suboptimal performance. This is demonstrated with the next example. √ √ Example 6. Consider the scenario with A = B source symbols of constant length L = L = B. From the preceding discussion, we see that this is the worst-case situation for variable-length deduplication, in which we can expect to see all possible A2 different boundary chunks. A slightly tightened version of Theorem 2 (which omits the last te(cid:0)rm in the denom(cid:1) inator using (cid:0)that L(cid:1)is constant) together with a lower bound derived in Appendix , show that then − ≤ R ≤ Ω B1/4 log 2(B) VL O B1/4 R(cid:63) → ∞ as B . This has two implications: First, that Theorem 2 is tight to within a polylogarithmic factor in ♦ B for this setting; and second that variable-length deduplication can still be polynomially suboptimal. The multi-chunk deduplication scheme proposed in this paper circumvents this trade-off by encoding multiple chunks jointly. This allows to choose the expected chunk length to be quite small, thereby limiting the effect of the boundary chunks, without the penalty of increased number of dictionary pointers. The next theorem bounds the performance of this scheme. Theorem 3. Consider the source model with B source blocks drawn with replacement from the A source symbols of expected length E(cid:18)(L). The performance of the mu(cid:0)lti-chunk d(cid:1)eduplication scheme with(cid:19)optimized marker length M satisﬁes the(cid:0)n (cid:1) E ≤ R ≤ B log AB (L) MC 1 R(cid:63) 1 + O min{A, B}(E(L) − 1) + (B − A)+ log(A/2) − 2B log(2E(L)) + → ∞ as B . (cid:0) (cid:1) The proof of Theorem 3 is reported in Appendix . The theorem shows that, under the fairly mild ≤ ≤ − E ≤ conditions BΩ(1) A (1 ε)B and (L) Aε/3 for some constant ε > 0, multi-chunk deduplication → ∞ ≤ ≤ E E is within a constant factor of optimal as B . Further, if A B o A (L)/ log(A (L)) , then → ∞ multi-chunk deduplication is asymptotically optimal as B . Example 4 (Continued). Consider again the scenario with A = 105 source symbols and with B = 106 E source blocks with expected length (L) = 106 bits. Theorem 3 (using the explicit constant in the order notation from Appendix ) shows then that multi-chunk deduplication has performance satisfying ≤ ≤ R(cid:63) R 1.2R(cid:63). MC By numerically optimizing the marker length M, this factor can be further reduced to 1.05. Thus, the ♦ proposed multi-chunk deduplication scheme is quite close to optimal in this setting. √ Example 6√(Continued). Consider again the scenario with A = B source symbols of constant length L = L = B. Recall that the variable-length deduplication scheme had a rate at least polynomially suboptimal: R ≥ VL Ω(B1/5). R(cid:63) On the other hand, a slightly tightened version of Theorem 3, which omits the last term in the denominator using that L is constant, shows that the rate of the multi-chunk deduplication scheme satisﬁes R ≤ MC O(1) R(cid:63) → ∞ as B . Thus, multi-chunk deduplication is order optimal in this case, as opposed to the polynomial ♦ loss factor of variable-length deduplication. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Unfortunately,

text: even

text: with

text: this

text: optimal

text: choice

text: of

text: marker

text: length,

text: using

text: deduplication

text: chunks

text: that

text: have

text: lengths

text: of

text: different

text: order

text: than

text: the

text: source

text: blocks

text: can

text: lead

text: to

text: asymptotically

text: signiﬁcantly

text: suboptimal

text: performance.

text: This

text: is

text: demonstrated

text: with

text: the

text: next

text: example.

text: Example

text: Consider

text: the

text: scenario

text: with

text: A

text: B

text: source

text: symbols

text: of

text: constant

text: length

text: L

text: L

text: B.

text: From

text: the

text: preceding

text: discussion,

text: we

text: see

text: that

text: this

text: is

text: the

text: worst-case

text: situation

text: for

text: variable-length

text: deduplication,

text: in

text: which

text: we

text: can

text: expect

text: to

text: see

text: all

text: possible

text: different

text: boundary

text: chunks.

text: A

text: slightly

text: tightened

text: version

text: of

text: Theorem

text: (which

text: omits

text: the

text: last

text: in

text: the

text: inator

text: using

text: constant)

text: together

text: with

text: a

text: lower

text: bound

text: derived

text: in

text: Appendix

text: show

text: that

text: then

text: R

text: Ω

text: log

text: VL

text: O

text: as

text: B

text: This

text: has

text: two

text: implications:

text: First,

text: that

text: Theorem

text: is

text: tight

text: to

text: within

text: a

text: polylogarithmic

text: factor

text: in

text: B

text: for

text: this

text: setting;

text: and

text: second

text: that

text: variable-length

text: deduplication

text: can

text: still

text: be

text: polynomially

text: suboptimal.

text: The

text: multi-chunk

text: deduplication

text: scheme

text: proposed

text: in

text: this

text: paper

text: circumvents

text: this

text: trade-off

text: by

text: encoding

text: multiple

text: chunks

text: jointly.

text: This

text: allows

text: to

text: choose

text: the

text: expected

text: chunk

text: length

text: to

text: be

text: quite

text: small,

text: thereby

text: limiting

text: the

text: effect

text: of

text: the

text: boundary

text: chunks,

text: without

text: the

text: penalty

text: of

text: increased

text: number

text: of

text: dictionary

text: pointers.

text: The

text: next

text: theorem

text: bounds

text: the

text: performance

text: of

text: this

text: scheme.

text: Theorem

text: Consider

text: the

text: source

text: model

text: with

text: B

text: source

text: blocks

text: drawn

text: with

text: replacement

text: from

text: the

text: A

text: source

text: symbols

text: of

text: expected

text: length

text: The

text: performance

text: of

text: the

text: scheme

text: marker

text: length

text: M

text: satisﬁes

text: E

text: R

text: B

text: log

text: AB

text: (L)

text: MC

text: O

text: min{A,

text: B}(E(L)

text: (B

text: A)+

text: as

text: B

text: The

text: proof

text: of

text: Theorem

text: is

text: reported

text: in

text: Appendix

text: The

text: theorem

text: shows

text: that,

text: under

text: the

text: fairly

text: mild

text: E

text: conditions

text: A

text: ε)B

text: and

text: (L)

text: for

text: some

text: constant

text: ε

text: multi-chunk

text: deduplication

text: E

text: E

text: is

text: within

text: a

text: constant

text: factor

text: of

text: optimal

text: as

text: B

text: Further,

text: if

text: A

text: B

text: o

text: A

text: (L)/

text: log(A

text: (L))

text: then

text: multi-chunk

text: deduplication

text: is

text: asymptotically

text: optimal

text: as

text: B

text: Example

text: (Continued).

text: Consider

text: again

text: the

text: scenario

text: with

text: A

text: source

text: symbols

text: and

text: with

text: B

text: E

text: source

text: blocks

text: with

text: expected

text: length

text: (L)

text: bits.

text: Theorem

text: (using

text: the

text: explicit

text: constant

text: in

text: the

text: order

text: notation

text: from

text: Appendix

text: shows

text: then

text: that

text: multi-chunk

text: deduplication

text: has

text: performance

text: satisfying

text: R

text: MC

text: By

text: numerically

text: optimizing

text: the

text: marker

text: length

text: M,

text: this

text: factor

text: can

text: be

text: further

text: reduced

text: to

text: Thus,

text: the

text: proposed

text: multi-chunk

text: deduplication

text: scheme

text: is

text: quite

text: close

text: to

text: optimal

text: in

text: this

text: setting.

text: Example

text: Consider

text: again

text: the

text: scenario

text: with

text: A

text: B

text: source

text: symbols

text: of

text: constant

text: length

text: L

text: L

text: B.

text: Recall

text: that

text: the

text: variable-length

text: deduplication

text: scheme

text: had

text: a

text: rate

text: at

text: least

text: polynomially

text: suboptimal:

text: R

text: VL

text: On

text: the

text: other

text: hand,

text: a

text: slightly

text: tightened

text: version

text: of

text: Theorem

text: which

text: omits

text: the

text: last

text: term

text: in

text: the

text: denominator

text: using

text: that

text: L

text: is

text: constant,

text: shows

text: that

text: the

text: rate

text: of

text: the

text: multi-chunk

text: deduplication

text: scheme

text: satisﬁes

text: R

text: MC

text: as

text: B

text: Thus,

text: multi-chunk

text: deduplication

text: is

text: order

text: optimal

text: in

text: this

text: case,

text: as

text: opposed

text: to

text: the

text: polynomial

text: loss

text: factor

text: of

text: variable-length

text: deduplication.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 (cid:0) (cid:12) (cid:1) (cid:12) (cid:0) (cid:12) (cid:1) (cid:0) (cid:12) (cid:1) (cid:12) (cid:12) | − ≥ − H(Y Yb 1) H Y Yb 1, I{ ∈Y − } b b Y b 1 P ∈ Y − b − ∈ Y − P ∈ Y − − ∈ Y − = (Y / b 1)H Y Yb 1, Y / b 1 + (Y b 1)H Y Yb 1, Y b 1 . (5) b b b b b b ∈ Y − − { } \ Y − Conditioned on Y /(cid:0) b (cid:12)1 and Yb 1, the s(cid:1)ource(cid:0)block Y is uniforml(cid:12)y distributed(cid:1) over 0, 1 L b 1. b (cid:12) b (cid:12) Hence, (cid:0) (cid:12) (cid:1) (cid:12) − ∈ Y − E |{ } \ Y − | ∈ Y − H Y Yb 1, Y / b 1 = log 0, 1 L b 1 Y / b 1 b b b E − |Y − | ∈ Y − = log(2L b 1 ) Y / b 1 b (≥a) − log(2L A) (≥b) − L 1 |Y − | ≤ (cid:0) (cid:12) ≤ − (cid:1) using that b 1 A for (a) and that A(cid:12) 2L 1 by assumption for (b). This implies that P ∈ Y − − ∈ Y − ≥ P ∈ Y − − (Y / b 1)H Y Yb 1, Y / b 1 (Y / b 1)(L 1). (6) b b b b ∈ Y − − Y − Conditioned on Y(cid:0) (cid:12) b 1 and Yb 1, th(cid:1)e source block Y is uniformly distributed over b 1. Hence, b (cid:12) b (cid:0) (cid:12) (cid:1) (cid:12) P ∈ Y − − ∈ Y − (Y b 1)H Y Yb 1, Y b 1 b b b P(cid:88) ∈ Y − − |Y − | ∈ Y − (cid:0) (cid:12) (cid:1) = (Y b 1)H Y Yb 1, b 1 , Y b 1 (cid:12) b b b A P |Y − | P ∈ Y − | |Y − | − |Y − | ∈ Y − = (cid:88) ( b 1 = a) (Y b 1 b 1 = a)H Y Yb 1, b 1 = a, Y b 1 b b b a=1 A P(cid:0)|Y − | − (cid:1) = ( b 1 = a)aA 1 log(a) a=1 − E |Y − | |Y − | = A 1 b 1 log b 1 . (7) Combining (3)–(7) yields(cid:88)(cid:16) (cid:0) (cid:1)(cid:17) ≥ B − P ∈ Y − − E |Y − | |Y − | R(cid:63) (L 1) (Y / b 1) + A 1 b 1 log b 1 . (8) b b=1 To obtain a m(cid:88)or(cid:16)e explicit expression, we further lower bound R(cid:63) as (cid:17) (≥a) B − − − − E|Y − | E|Y − | R(cid:63) (L (cid:88)1)(1 (b 1)/A)+ +(cid:88)A (cid:0)1 b 1 log b(cid:1)1 (cid:0) (cid:0) (cid:1)(cid:1) b=1 − B − − B − − − − − − (=b) (L 1) (1 (b 1)/A)+ + 1 (1 1/A)b 1 log A 1 (1 1/A)b 1 b=1 b=1 (≥c) − { } − 0.5(L 1) min A, B + 0.5(B A)+ log(A/2), (9)

label: paragraph

label: equation

text: H(Y

text: Yb

text: H

text: Y

text: Yb

text: I{

text: ∈Y

text: b

text: b

text: Y

text: b

text: P

text: Y

text: b

text: Y

text: P

text: Y

text: Y

text: (Y

text: b

text: Y

text: Yb

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text: Yb

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text: b

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text: b

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text: Y

text: Conditioned

text: on

text: Y

text: b

text: and

text: Yb

text: the

text: Y

text: is

text: over

text: L

text: b

text: b

text: b

text: Hence,

text: Y

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text: Y

text: Y

text: H

text: Y

text: Yb

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text: L

text: b

text: Y

text: b

text: b

text: b

text: b

text: E

text: |Y

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text: b

text: Y

text: b

text: b

text: (≥a)

text: A)

text: (≥b)

text: L

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text: that

text: b

text: A

text: for

text: (a)

text: and

text: that

text: by

text: assumption

text: for

text: (b).

text: This

text: implies

text: that

text: P

text: Y

text: Y

text: P

text: Y

text: (Y

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text: Y

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text: on

text: b

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text: Yb

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text: Y

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text: uniformly

text: distributed

text: over

text: b

text: Hence,

text: b

text: b

text: P

text: Y

text: Y

text: (Y

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text: Y

text: Yb

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text: b

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text: Y

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text: A

text: P

text: |Y

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text: Y

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text: (Y

text: b

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text: a)H

text: Y

text: Yb

text: b

text: a,

text: Y

text: b

text: b

text: b

text: b

text: A

text: b

text: a)aA

text: log(a)

text: E

text: |Y

text: |Y

text: A

text: b

text: log

text: b

text: Combining

text: B

text: P

text: Y

text: E

text: |Y

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text: (L

text: (Y

text: b

text: A

text: b

text: log

text: b

text: b

text: To

text: obtain

text: a

text: explicit

text: expression,

text: we

text: further

text: lower

text: bound

text: as

text: (≥a)

text: B

text: E|Y

text: E|Y

text: (L

text: (b

text: b

text: log

text: B

text: B

text: (=b)

text: (L

text: (b

text: log

text: A

text: (≥c)

text: min

text: A,

text: B

text: A)+


text: Pairwise velocities in the “Running FLRW” cosmological model 11 Figure 4. The ratio of the pairwise velocity, and the dispersions, σ|| and σ12 , between the CoDECS simulations and the main ΛCDM run at ﬁxed separation r = 5 Mpc/h at redshift z = 0. The error bars show the 1-σ uncertainty calculated using a bootstrap technique. Each statistic is also calculated for three additional ΛCDM runs with the same σ8 value at redshift z = 0 as the CoDECS scenarios EXP002, EXP003 and EXP008e3, represented as colored bands enclosing the 1-σ region around each measurement. baryons and dark matter, they point to the absence of a use the latest determination of the cosmological parameters, velocity bias. This means that velocities measured from ha- and in their analysis we focused on the degeneracies that los in cosmological simulations can be readily compared to these observables present. We also extended our analysis to galaxy observations without further adjustments, and so a second suite of simulations built on diﬀerent dark energy without the uncertainties that such adjustments always in- models, increasing the number of scenarios with pairwise ve- clude. locity predictions. By doing so we highlighted very impor- 6 SUMMARY tant deviations which have the potential to be important “smoking guns” for some Coupled Dark Energy models. In The aim of this work was to contribute to the current fact, these modiﬁcations inﬂuence the velocity statistics in a eﬀorts being undertaken towards the exploitation of velocity way that is not degenerate with σ , meaning that the results 8 statistics as probes of modiﬁed gravity and dark energy be- presented here, in combination with future survey data, will haviours. In particular, we limited the scope of our study to help to further test and possibly falsify these scenarios. Our pairwise velocities as these were recently shown to provide results warrant further theoretical study to better under- interesting signatures for modiﬁed gravity models. To this stand the mechanisms that lead to such diﬀerences in these end we analyzed the velocities from a suite of N-body sim- models. ulations in the R-FLRW cosmological model, extending the ACKNOWLEDGMENTS current literature and providing tests against recent cosmo- logical observations and for future surveys. Our simulations We would like to thank Manodeep Sinha for making © 0000 RAS, MNRAS 000, 000–000 ##LTFigure##

text: Pairwise

text: velocities

text: in

text: the

text: “Running

text: FLRW”

text: cosmological

text: model

text: Figure

text: The

label: caption

text: ratio

text: of

text: the

text: pairwise

text: velocity,

text: and

text: the

text: dispersions,

text: σ||

text: and

text: between

text: the

text: CoDECS

text: simulations

text: and

text: the

text: main

text: ΛCDM

text: run

text: at

text: ﬁxed

text: separation

text: r

text: Mpc/h

text: at

text: redshift

text: z

text: The

text: error

text: bars

text: show

text: the

text: uncertainty

text: calculated

text: using

text: a

text: bootstrap

text: technique.

text: Each

text: statistic

text: is

text: also

text: calculated

text: for

text: three

text: additional

text: ΛCDM

text: runs

text: with

text: the

text: same

text: σ8

text: value

text: at

text: redshift

text: z

text: as

text: the

text: CoDECS

text: scenarios

text: and

text: represented

text: as

text: colored

text: bands

text: enclosing

text: the

text: region

text: around

text: each

text: measurement.

text: baryons

text: and

text: dark

text: matter,

text: they

text: point

text: to

text: the

text: absence

text: of

text: a

text: use

text: the

text: latest

text: determination

text: of

text: the

text: cosmological

text: parameters,

text: velocity

text: bias.

text: This

text: means

text: that

text: velocities

text: measured

text: from

text: ha-

text: and

text: in

text: their

text: analysis

text: we

text: focused

text: on

text: the

text: degeneracies

text: that

text: los

text: in

text: cosmological

text: simulations

text: can

text: be

text: readily

text: compared

text: to

text: these

text: observables

text: present.

text: We

text: also

text: extended

text: our

text: analysis

text: to

text: galaxy

text: observations

text: without

text: further

text: adjustments,

text: and

text: so

text: a

text: second

text: suite

text: of

text: simulations

text: built

text: on

text: diﬀerent

text: dark

text: energy

text: without

text: the

text: uncertainties

text: that

text: such

text: adjustments

text: always

text: in-

text: models,

text: increasing

text: the

text: number

text: of

text: scenarios

text: with

text: pairwise

text: ve-

text: clude.

text: locity

text: predictions.

text: By

text: doing

text: so

text: we

text: highlighted

text: very

text: impor-

text: SUMMARY

text: tant

text: deviations

text: which

text: have

text: the

text: potential

text: to

text: be

text: important

text: “smoking

text: guns”

text: for

text: some

text: Coupled

text: Dark

text: Energy

text: models.

text: In

text: The

text: aim

text: of

text: this

text: work

text: was

text: to

text: contribute

text: to

text: the

text: current

text: fact,

text: these

text: modiﬁcations

text: inﬂuence

text: the

text: velocity

text: statistics

text: in

text: a

text: eﬀorts

text: being

text: undertaken

text: towards

text: the

text: exploitation

text: of

text: velocity

text: way

text: that

text: is

text: not

text: degenerate

text: with

text: σ

text: meaning

text: that

text: the

text: results

text: statistics

text: as

text: probes

text: of

text: modiﬁed

text: gravity

text: and

text: dark

text: energy

text: be-

text: presented

text: here,

text: in

text: combination

text: with

text: future

text: survey

text: data,

text: will

text: haviours.

text: In

text: particular,

text: we

text: limited

text: the

text: scope

text: of

text: our

text: study

text: to

text: help

text: to

text: further

text: test

text: and

text: possibly

text: falsify

text: these

text: scenarios.

text: Our

text: pairwise

text: velocities

text: as

text: these

text: were

text: recently

text: shown

text: to

text: provide

text: results

text: warrant

text: further

text: theoretical

text: study

text: to

text: better

text: under-

text: interesting

text: signatures

text: for

text: modiﬁed

text: gravity

text: models.

text: To

text: this

text: stand

text: the

text: mechanisms

text: that

text: lead

text: to

text: such

text: diﬀerences

text: in

text: these

text: end

text: we

text: analyzed

text: the

text: velocities

text: from

text: a

text: suite

text: of

text: N-body

text: sim-

text: models.

text: ulations

text: in

text: the

text: R-FLRW

text: cosmological

text: model,

text: extending

text: the

text: ACKNOWLEDGMENTS

text: current

text: literature

text: and

text: providing

text: tests

text: against

text: recent

text: cosmo-

text: logical

text: observations

text: and

text: for

text: future

text: surveys.

text: Our

text: simulations

text: We

text: would

text: like

text: to

text: thank

text: Manodeep

text: Sinha

text: for

text: making

text: RAS,

text: MNRAS

text: ##LTFigure##


text: Pairwise velocities in the “Running FLRW” cosmological model 9 Simulation σ8(z = 0) of the coupling, and within the same model by the strength of the coupling. The names of the various runs, along with ΛCDM 0.809 the value of σ at redshift z = 0, are reported in Table 3: the EXP001 0.825 8 EXP001, EXP002 and EXP003 runs simulate an exponential EXP002 0.875 potential (Wetterich 1988) and a constant coupling, with re- EXP003 0.967 EXP008e3 0.895 spectively increasing strength; the EXP008e3 run simulates SUGRA003 0.806 an exponential potential with a variable coupling; and the SUGRA003 run simulates a supergravity potential (Brax & Table 3. The diﬀerent CoDECS simulations distinguished by the Martin 1999) with a constant coupling. The details of the shape of the potential and the functional form and/or strength new dynamics and their background evolution is outside the of the coupling between the dark energy ﬁeld and dark matter. scope of this work but is extensively reviewed in the in- All the models have the same amplitude of matter perturbations troductory paper of the simulations suite, Baldi (2012), to at the redshift of the CMB the linear perturbations but evolve in which we refer the interested reader. a very diﬀerent way, this results in very diﬀerent values of σ8 at redshift z = 0, reported in the last column. While the additional parameters introduced by the dif- ferent CoDECS models vary between the simulations, the base cosmological parameters are the same across all the runs and agree with the “WMAP7 only Maximum Likeli- shown in Figure 1 where we investigated the the dark matter hood” results by Komatsu et al. (2011). The CoDECS suite distribution. With our models it is only possible to break was also the ﬁrst to adopt a clear convention about the nor- this degeneracy by probing fully non-linear scales and their malization of the initial conditions: all the models simulated redshift evolution, as we have shown in Bibiano & Croton agree on the measure of σ at the redshift of recombina- (2016). ∼ 8 tion, z 1100. This is the same convention we adopt for 5 DISCUSSION our simulations, and for all the models in this suite, except Thus far we have focused on the statistics of pairwise the SUGRA scenario, leads to a very diﬀerent σ8 at red- velocities to investigate how they are sensitive to the running shift z = 0. For this reason we will focus our investigation of fundamental parameters introduced in the R-FLRW cos- of the pairwise velocity statistics on the degeneracies with mology. This was motivated by the recent ﬁndings from Hell- the value of σ8, as before. wing et al. (2014) and Fagernes Ivarsen et al. (2016), where All the ﬁve simulations in CoDECS follow the evolu- σ|| and σ were found to be particularly sensitive to modi- tion of 10243 dark matter particles and 10243 baryon parti- 12 ﬁed gravity eﬀects. We developed our own high performance cles in a periodic box 1 Gpc/h aside. The baryon particles pairwise statistics code, and once run on our R-FLRW sim- are treated as collisionless and do not receive the usual hy- ulations found that such velocity statistics are indeed mod- drodynamic treatment. The necessity to include them as a iﬁed by the new dynamics. In particular, the dispersions in separate kind of particle stems from the nature of the mod- the R-FLRW scenarios show excesses of up to 10% when els, as the dark matter particles feel the additional ﬁfth force compared to ΛCDM. Knowing that these excesses are usu- — which is not felt by the baryons — due to the coupling ally degenerate with σ we then ran additional simulations being limited to the dark sector. Additionally, there is an 8 to quantify this. This revealed that the degree of degener- energy-momentum transfer between the dark energy ﬁeld acy is signiﬁcant. For example, the redshift z = 0 pairwise and the dark matter ﬂuid which results in an evolution of velocity of halos in a R-FLRW scenario and a ΛCDM uni- the dark matter particle mass with time. Aside from these verse with the same present-day σ are indistinguishable to additional details the main simulation algorithms and prod- 8 within the errors of our simulations. Unfortunately exploit- ucts are very similar to our simulations suite: the CoDECS ing the redshift evolution of these observables does not help, N-body code is a modiﬁed version of the gadget code, and as even at higher redshifts we fail to register any signiﬁcant in post processing the FOF and SUBFIND algorithms were diﬀerence. used to identify halos and sub-halos. This σ degeneracy was not studied previously in the lit- To begin, we run a partner set of ΛCDM simulations 8 erature when considering such observables. In the R-FLRW having the same ﬁnal σ8 of each CoDECS simulation, as case it mostly results from the nature of the modiﬁcation before. We then adapt our analysis pipeline for the CoDECS introduced by the model, having a time dependance but no simulation suite, splitting the full catalog into diﬀerent mass scale dependence. bins and measuring the previous velocity statistics on all To understand if this is a result shared by other dark simulations: v12 and the dispersions σ|| and σ12, as well as energy models also making time dependent modiﬁcations performing the same error analysis described in Section . to the standard cosmology we extend our analysis pipeline In Figure 4 we present the results of this, where we to examine the “Coupled dark energy Cosmological Simu- ﬁnd the same trends seen in the R-FLRW simulations, with lations” (CoDECS) suite by Baldi (2012). In such coupled both a mass and separation dependence. Here we focus on dark energy models the dark energy component of the Uni- the magnitude of the dispersions which reveal some inter- verse is treated as a scalar ﬁeld which evolves due to a po- esting deviations for two CoDECS simulations in particular, tential, but also interacts with the dark matter ﬂuid. This EXP003 and EXP008e3. These exhibit an excesses of be- interaction results in a ﬁfth force, felt only by dark matter tween 15% and 20% in the pairwise velocity and between particles, which inﬂuences the formation of structures both 20% and 30% in both dispersion measurements. The other in the linear and non-linear regime. For the CoDECS suite simulations show smaller deviations from ΛCDM, with the the models chosen are distinguished by the choice of poten- SUGRA model being the only one to exhibit smaller veloc- tial for the dark energy scalar ﬁeld, by the functional forms ities and dispersion than the reference ΛCDM simulation. © 0000 RAS, MNRAS 000, 000–000 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Pairwise

text: velocities

text: in

text: the

text: “Running

text: FLRW”

text: cosmological

text: model

text: Simulation

label: table

text: of

text: the

text: coupling,

text: and

text: within

text: the

text: same

text: model

text: by

text: the

text: strength

text: of

text: the

text: coupling.

text: The

text: names

text: of

text: the

text: various

text: runs,

text: along

text: with

text: ΛCDM

text: the

text: value

text: of

text: σ

text: at

text: redshift

text: z

text: are

text: reported

text: in

text: Table

text: the

text: and

text: runs

text: simulate

text: an

text: exponential

text: potential

text: (Wetterich

text: and

text: a

text: constant

text: coupling,

text: with

text: re-

text: spectively

text: increasing

text: strength;

text: the

text: run

text: simulates

text: an

text: exponential

text: potential

text: with

text: a

text: variable

text: coupling;

text: and

text: the

text: run

text: simulates

text: a

text: supergravity

text: potential

text: (Brax

text: Table

text: The

text: diﬀerent

text: CoDECS

text: simulations

text: distinguished

text: by

text: the

text: Martin

text: with

text: a

text: constant

text: coupling.

text: The

text: details

text: of

text: the

text: shape

text: of

text: the

text: potential

text: and

text: the

text: functional

text: form

text: and/or

text: strength

text: new

text: dynamics

text: and

text: their

text: background

text: evolution

text: is

text: outside

text: the

text: of

text: the

text: coupling

text: between

text: the

text: dark

text: energy

text: ﬁeld

text: and

text: dark

text: matter.

text: scope

text: of

text: this

text: work

text: but

text: is

text: extensively

text: reviewed

text: in

text: the

text: in-

text: All

text: the

text: models

text: have

text: the

text: same

text: amplitude

text: of

text: matter

text: perturbations

text: troductory

text: paper

text: of

text: the

text: simulations

text: suite,

text: Baldi

text: to

text: at

text: the

text: redshift

text: of

text: the

text: CMB

text: the

text: linear

text: perturbations

text: but

text: evolve

text: in

text: which

text: we

text: refer

text: the

text: interested

text: reader.

text: a

text: very

text: diﬀerent

text: way,

text: this

text: results

text: in

text: very

text: diﬀerent

text: values

text: of

text: σ8

text: at

text: redshift

text: z

text: reported

text: in

text: the

text: last

text: column.

text: While

text: the

text: additional

text: parameters

text: introduced

text: by

text: the

text: dif-

text: ferent

text: CoDECS

text: models

text: vary

text: between

text: the

text: simulations,

text: the

text: base

text: cosmological

text: parameters

text: are

text: the

text: same

text: across

text: all

text: the

text: runs

text: and

text: agree

text: with

text: the

text: only

text: Maximum

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text: shown

text: in

text: Figure

text: where

text: we

text: investigated

text: the

text: the

text: dark

text: matter

text: hood”

text: results

text: by

text: Komatsu

text: et

text: al.

text: The

text: CoDECS

text: suite

text: distribution.

text: With

text: our

text: models

text: it

text: is

text: only

text: possible

text: to

text: break

text: was

text: also

text: the

text: ﬁrst

text: to

text: adopt

text: a

text: clear

text: convention

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text: the

text: nor-

text: this

text: degeneracy

text: by

text: probing

text: fully

text: non-linear

text: scales

text: and

text: their

text: malization

text: of

text: the

text: initial

text: conditions:

text: all

text: the

text: models

text: simulated

text: redshift

text: evolution,

text: as

text: we

text: have

text: shown

text: in

text: Bibiano

text: Croton

text: agree

text: on

text: the

text: measure

text: of

text: σ

text: at

text: the

text: redshift

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text: tion,

text: z

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text: same

text: convention

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text: adopt

text: for

text: DISCUSSION

text: our

text: simulations,

text: and

text: for

text: all

text: the

text: models

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text: this

text: suite,

text: except

text: Thus

text: far

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text: have

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text: on

text: the

text: statistics

text: of

text: pairwise

text: the

text: SUGRA

text: scenario,

text: leads

text: to

text: a

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text: diﬀerent

text: σ8

text: at

text: red-

text: velocities

text: to

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text: how

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text: are

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text: shift

text: z

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text: our

text: investigation

text: of

text: fundamental

text: parameters

text: introduced

text: in

text: the

text: R-FLRW

text: cos-

text: of

text: the

text: pairwise

text: velocity

text: statistics

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text: the

text: degeneracies

text: with

text: mology.

text: This

text: was

text: motivated

text: by

text: the

text: recent

text: ﬁndings

text: from

text: Hell-

text: the

text: value

text: of

text: as

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text: wing

text: et

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text: Fagernes

text: Ivarsen

text: et

text: al.

text: where

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text: simulations

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text: CoDECS

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text: evolu-

text: σ||

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text: σ

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text: tion

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text: ﬁed

text: gravity

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text: We

text: developed

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text: aside.

text: The

text: baryon

text: particles

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text: statistics

text: code,

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text: on

text: our

text: R-FLRW

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text: are

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text: ulations

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text: velocity

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text: mod-

text: drodynamic

text: treatment.

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text: necessity

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text: particular,

text: the

text: dispersions

text: in

text: separate

text: kind

text: of

text: particle

text: stems

text: from

text: the

text: nature

text: of

text: the

text: mod-

text: the

text: R-FLRW

text: scenarios

text: show

text: excesses

text: of

text: up

text: to

text: when

text: els,

text: as

text: the

text: dark

text: matter

text: particles

text: feel

text: the

text: additional

text: ﬁfth

text: force

text: compared

text: to

text: ΛCDM.

text: Knowing

text: that

text: these

text: excesses

text: are

text: usu-

text: which

text: is

text: not

text: felt

text: by

text: the

text: baryons

text: due

text: to

text: the

text: coupling

text: ally

text: degenerate

text: with

text: σ

text: we

text: then

text: ran

text: additional

text: simulations

text: being

text: limited

text: to

text: the

text: dark

text: sector.

text: Additionally,

text: there

text: is

text: an

text: to

text: quantify

text: this.

text: This

text: revealed

text: that

text: the

text: degree

text: of

text: degener-

text: energy-momentum

text: transfer

text: between

text: the

text: dark

text: energy

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text: acy

text: is

text: signiﬁcant.

text: For

text: example,

text: the

text: redshift

text: z

text: pairwise

text: and

text: the

text: dark

text: matter

text: ﬂuid

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text: results

text: in

text: an

text: evolution

text: of

text: velocity

text: of

text: halos

text: in

text: a

text: R-FLRW

text: scenario

text: and

text: a

text: ΛCDM

text: uni-

text: the

text: dark

text: matter

text: particle

text: mass

text: with

text: time.

text: Aside

text: from

text: these

text: verse

text: with

text: the

text: same

text: present-day

text: σ

text: are

text: indistinguishable

text: to

text: additional

text: details

text: the

text: main

text: simulation

text: algorithms

text: and

text: prod-

text: within

text: the

text: errors

text: of

text: our

text: simulations.

text: Unfortunately

text: exploit-

text: ucts

text: are

text: very

text: similar

text: to

text: our

text: simulations

text: suite:

text: the

text: CoDECS

text: ing

text: the

text: redshift

text: evolution

text: of

text: these

text: observables

text: does

text: not

text: help,

text: N-body

text: code

text: is

text: a

text: modiﬁed

text: version

text: of

text: the

text: gadget

text: code,

text: and

text: as

text: even

text: at

text: higher

text: redshifts

text: we

text: fail

text: to

text: register

text: any

text: signiﬁcant

text: in

text: post

text: processing

text: the

text: FOF

text: and

text: SUBFIND

text: algorithms

text: were

text: diﬀerence.

text: used

text: to

text: identify

text: halos

text: and

text: sub-halos.

text: This

text: σ

text: degeneracy

text: was

text: not

text: studied

text: previously

text: in

text: the

text: lit-

text: To

text: begin,

text: we

text: run

text: a

text: partner

text: set

text: of

text: ΛCDM

text: simulations

text: erature

text: when

text: considering

text: such

text: observables.

text: In

text: the

text: R-FLRW

text: having

text: the

text: same

text: ﬁnal

text: σ8

text: of

text: each

text: CoDECS

text: simulation,

text: as

text: case

text: it

text: mostly

text: results

text: from

text: the

text: nature

text: of

text: the

text: modiﬁcation

text: before.

text: We

text: then

text: adapt

text: our

text: analysis

text: pipeline

text: for

text: the

text: CoDECS

text: introduced

text: by

text: the

text: model,

text: having

text: a

text: time

text: dependance

text: but

text: no

text: simulation

text: suite,

text: splitting

text: the

text: full

text: catalog

text: into

text: diﬀerent

text: mass

text: scale

text: dependence.

text: bins

text: and

text: measuring

text: the

text: previous

text: velocity

text: statistics

text: on

text: all

text: To

text: understand

text: if

text: this

text: is

text: a

text: result

text: shared

text: by

text: other

text: dark

text: simulations:

text: and

text: the

text: dispersions

text: σ||

text: and

text: as

text: well

text: as

text: energy

text: models

text: also

text: making

text: time

text: dependent

text: modiﬁcations

text: performing

text: the

text: same

text: error

text: analysis

text: described

text: in

text: Section

text: to

text: the

text: standard

text: cosmology

text: we

text: extend

text: our

text: analysis

text: pipeline

text: In

text: Figure

text: we

text: present

text: the

text: results

text: of

text: this,

text: where

text: we

text: to

text: examine

text: the

text: “Coupled

text: dark

text: energy

text: Cosmological

text: Simu-

text: ﬁnd

text: the

text: same

text: trends

text: seen

text: in

text: the

text: R-FLRW

text: simulations,

text: with

text: lations”

text: (CoDECS)

text: suite

text: by

text: Baldi

text: In

text: such

text: coupled

text: both

text: a

text: mass

text: and

text: separation

text: dependence.

text: Here

text: we

text: focus

text: on

text: dark

text: energy

text: models

text: the

text: dark

text: energy

text: component

text: of

text: the

text: Uni-

text: the

text: magnitude

text: of

text: the

text: dispersions

text: which

text: reveal

text: some

text: inter-

text: verse

text: is

text: treated

text: as

text: a

text: scalar

text: ﬁeld

text: which

text: evolves

text: due

text: to

text: a

text: po-

text: esting

text: deviations

text: for

text: two

text: CoDECS

text: simulations

text: in

text: particular,

text: tential,

text: but

text: also

text: interacts

text: with

text: the

text: dark

text: matter

text: ﬂuid.

text: This

text: and

text: These

text: exhibit

text: an

text: excesses

text: of

text: be-

text: interaction

text: results

text: in

text: a

text: ﬁfth

text: force,

text: felt

text: only

text: by

text: dark

text: matter

text: tween

text: and

text: in

text: the

text: pairwise

text: velocity

text: and

text: between

text: particles,

text: which

text: inﬂuences

text: the

text: formation

text: of

text: structures

text: both

text: and

text: in

text: both

text: dispersion

text: measurements.

text: The

text: other

text: in

text: the

text: linear

text: and

text: non-linear

text: regime.

text: For

text: the

text: CoDECS

text: suite

text: simulations

text: show

text: smaller

text: deviations

text: from

text: ΛCDM,

text: with

text: the

text: the

text: models

text: chosen

text: are

text: distinguished

text: by

text: the

text: choice

text: of

text: poten-

text: SUGRA

text: model

text: being

text: the

text: only

text: one

text: to

text: exhibit

text: smaller

text: veloc-

text: tial

text: for

text: the

text: dark

text: energy

text: scalar

text: ﬁeld,

text: by

text: the

text: functional

text: forms

text: ities

text: and

text: dispersion

text: than

text: the

text: reference

text: ΛCDM

text: simulation.

text: RAS,

text: MNRAS

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text: (cid:32) (cid:33) (cid:18) (cid:19) Hence using the above inequality, (7) and (6) we get − √ 1 P −3 p n ≤ − ψ < e k + σ2 exp ( n) , 2 2 k and the proof of (4) is complete. We now turn to proving the upper bound part of Theorem 3.1. We begin by establishing several preliminary results. 3.2 Preliminaries ≤ ≤ ≤ We ﬁrst observe that k log k Cn and n k log p/(2 log D ), implies log k C log p/(2 log D ). 0 0 In particular, for D suﬃciently large 0 ≤ k4 p. (8) We establish the following two auxiliary lemmas.√ ∈ N ≥ (cid:18) (cid:19) ≤ Lemma 3.2. If m , m with m 4 and m m then 1 2 1 2 1 m ≥ mm2 1 1 . m 4m ! 2 2 (cid:18) (cid:19) Proof. We have, m ≥ mm2 1 1 m 4m ! 2 2 (cid:18) (cid:19) holds if an only if (cid:89) − m2 1 − i ≥ 1 1 . m 4 1 i=1 √ ≤ (cid:18) (cid:19) (cid:18) (cid:19) Now m m implies (cid:89) (cid:89) 2 1 √ − (cid:98) (cid:99) m2 1 − i ≥ (cid:18) m1 − (cid:19)i 1 1 m m 1 1 i=1 i=1 √ ≥ − √1 m1 (cid:0) (cid:1) 1 , m 1 ≥ − ≥ It is easy to verify that x 2 implies 1 1 x 1. This completes the proof. x 4 16 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Hence

text: using

text: the

text: above

text: inequality,

text: and

text: we

text: get

text: P

text: −3

text: p

text: n

text: ψ

text: e

text: k

text: σ2

text: exp

text: n)

text: k

text: and

text: the

text: proof

text: of

text: is

text: complete.

text: We

text: now

text: turn

text: to

text: proving

text: the

text: upper

text: bound

text: part

text: of

text: Theorem

text: We

text: begin

text: by

text: establishing

text: several

text: preliminary

text: results.

label: section

text: Preliminaries

text: We

text: ﬁrst

text: observe

text: that

text: k

text: log

text: k

text: Cn

text: and

text: n

text: k

text: log

text: log

text: D

text: implies

text: log

text: k

text: C

text: log

text: log

text: D

text: In

text: particular,

text: for

text: D

text: suﬃciently

text: large

text: p.

text: We

text: establish

text: the

text: following

text: two

text: auxiliary

text: lemmas.√

text: N

text: Lemma

text: If

text: m

text: m

text: with

text: m

text: and

text: m

text: m

text: then

text: m

text: m

text: Proof.

text: We

text: have,

text: m

text: m

text: holds

text: if

text: an

text: only

text: if

text: i

text: m

text: Now

text: m

text: m

text: implies

text: i

text: m

text: m

text: √1

text: m

text: It

text: is

text: easy

text: to

text: verify

text: that

text: x

text: implies

text: x

text: This

text: completes

text: the

text: proof.

text: x

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text: ≥ Observe that if Υ 1 + ζn, then (12(cid:88)) imp(cid:16)lies (cid:17)that(cid:89)at least one of the summands of k p −l n qt,√Yi ,l 4 k k2 p2 (cid:96)=1 i=1 t,√Yi k for (cid:96) = 1, 2.., k should be at least ζn . Hence applying the union bound, k   P { − } ≥ ≤ P (cid:91)≥ (cid:16) (cid:17) (cid:89) (min 1, Υ 1 ζn) (Υ 1 + ζn)  ≤ P k{4 p −l n qt,√Yik,kl ≥ ζn} (cid:88) (cid:16) k2(cid:17) (cid:89) p2 k (cid:96)=1 i=1 t,√Yi  k ≤ k P p −l n qt,√Yi , l ≥ ζn 4 k k k2 p2 k (cid:96)=1 i=1 t,√Yi k (cid:88) Introducing parameter ρ = l we obtain k E { − } ≤ P { − } ≥ ≤ P (min 1, Υ 1 ) ζn + (min 1, Υ 1 ζn) ζn + (Υ ) , (13) Y ρ ρ=1,2,..,1 k k − (cid:110) (cid:16) (cid:17) (cid:89) (cid:111) where for all ρ = 1, .., k 1, k we deﬁne k k k (cid:44) p −ρk n qt,√Yi ,ρ ≥ ζn Υ 4 k . ρ k2 p2 k i=1 t,√Yi k P ∈ Next we obtain an upper bound on (Υ ) for any ρ (0, 1] as a function of ζ. Set ρ − n log D ρ∗ := 1 . 3k log(p/k2) ≤ The cases ρ ρ∗ and ρ > ρ∗ will be considered separately. (cid:16) (cid:17) ∈ ∈ Lemma 3.6. For all ρ (ρ∗, 1] and ζ (0, 1). P ≤ − − n (Υ ) 2n D 1 ζ 1 . ρ 18 6 (cid:0) (cid:1) Proof. Since ρ > ρ∗ then − − k log p ≥ −1 (1 ρ) k2 log D. (14) n 3 Now we have q ≤ p which implies qt,√Yik,ρ ≤ p−1 , which after taking logarithms and t,√Yik,ρ t,(cid:32)√Yik p2t,√Yi t,√Yik (cid:33) (cid:88) k dividing both the sides by n gives P ≤ P 1 n − ≥ − log 4k k log p (Υ ) log p log ζ + ρ k2 . ρ n t,√Yi n n k i=1 21 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Observe

text: that

text: if

text: Υ

text: ζn,

text: then

text: least

text: one

text: of

text: the

text: summands

text: of

text: k

text: p

text: −l

text: n

text: qt,√Yi

text: ,l

text: k

text: t,√Yi

text: k

text: for

text: k

text: should

text: be

text: at

text: least

text: ζn

text: Hence

text: applying

text: the

text: union

text: bound,

text: k

text: P

text: P

text: (min

text: Υ

text: ζn)

text: (Υ

text: ζn)

text: P

text: p

text: −l

text: n

text: qt,√Yik,kl

text: ζn}

text: k

text: t,√Yi

text: k

text: k

text: P

text: p

text: −l

text: n

text: qt,√Yi

text: l

text: ζn

text: k

text: k

text: k

text: t,√Yi

text: k

text: Introducing

text: parameter

text: ρ

text: l

text: we

text: obtain

text: k

text: E

text: P

text: P

text: (min

text: Υ

text: ζn

text: (min

text: Υ

text: ζn)

text: ζn

text: (Υ

text: Y

text: ρ

text: k

text: k

text: where

text: for

text: all

text: ρ

text: k

text: k

text: we

text: deﬁne

text: k

text: k

text: k

text: p

text: −ρk

text: n

text: qt,√Yi

text: ,ρ

text: ζn

text: Υ

text: k

text: ρ

text: k

text: t,√Yi

text: k

text: P

text: Next

text: we

text: obtain

text: an

text: upper

text: bound

text: on

text: (Υ

text: for

text: any

text: ρ

text: as

text: a

text: function

text: of

text: ζ.

text: Set

text: ρ

text: n

text: log

text: D

text: ρ∗

text: The

text: cases

text: ρ

text: ρ∗

text: and

text: ρ

text: ρ∗

text: will

text: be

text: considered

text: separately.

text: Lemma

text: For

text: all

text: ρ

text: (ρ∗,

text: and

text: ζ

text: P

text: n

text: (Υ

text: D

text: ζ

text: ρ

text: Proof.

text: Since

text: ρ

text: ρ∗

text: then

text: k

text: log

text: p

text: −1

text: ρ)

text: log

text: D.

text: n

text: Now

text: we

text: have

text: q

text: p

text: which

text: implies

text: qt,√Yik,ρ

text: which

text: after

text: taking

text: logarithms

text: and

text: t,√Yik,ρ

text: t,√Yik

text: k

text: dividing

text: both

text: the

text: sides

text: by

text: n

text: gives

text: P

text: P

text: n

text: log

text: k

text: log

text: p

text: (Υ

text: log

text: p

text: log

text: ζ

text: ρ

text: ρ

text: n

text: t,√Yi

text: n

text: n

text: k

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text: (cid:18) (cid:19) (cid:18) (cid:19) Combining the above with (29) we obtain that w.h.p., 2Λ + σ2 ≤ 2k −kk(−1−Λ(cid:15)p) c2 p + 1 , 2k + σ2 σ2 (cid:18) (cid:19) (cid:18) (cid:19) or w.h.p., 2Λ ≤ 2k −1−(cid:15) (cid:15)+k(Λ1−p(cid:15)) c2 p + 1 + 1 . (30) σ2 σ2 (cid:18) (cid:19) from which we obtain a simpler bound ≤ 2k −1−(cid:15) (cid:15)+k(Λ1−p(cid:15)) c2 + 1 , σ2 (cid:18) (cid:19) (cid:18) (cid:19) namely ≤ − (cid:15) Λ 2k 2 log c − + −p log + 1 1 (cid:15) k (1 (cid:15)) σ2 or (cid:0) (cid:1) 2 log c − ≤ Λ p (1 (cid:15)) + (cid:15) . log 2k + 1 k σ2 → ∞ ≥ Since by the assumption of the theorem we have k/σ2 , we obtain that Λp (cid:15)/2 w.h.p. as → ∞ (cid:18) (cid:19) (cid:18) (cid:19) k k . Now we reapply this bound for (30) and obtain that w.h.p. (cid:15)k ≤ 2k −1−(cid:15) (cid:15)+k(Λ1−p(cid:15)) c2 + 1 + 1 . σ2 σ2 (cid:18) (cid:19) (cid:18) (cid:18) (cid:19) (cid:19) Taking logarithm of both sides, we obtain that w.h.p. − − 2k (cid:15)k ≤ Λ (1 (cid:15)) log 1 + 1 log + 1 + 2 log c + (cid:15) p . σ2 σ2 k → ∞ Now again since k/σ2 , it is easy to see that the ratio of two logarithms approaches unity − as k increases, and thus the limit of the left-hand side is 1 (cid:15) + (cid:15) = 1 in the limit. Thus Λ /k p approaches unity in the limit w.h.p. as k increases. This completes the proof. 6 The Overlap Gap Property In this section we prove Theorem 2.5. We begin by establishing a certain property regarding the the limiting curve function Γ. Lemma 6.1. Under the assumption of Theor(cid:18)em 2.5, there(cid:19)exist sequences 0 < ζ < ζ < 1 − ∞ 1,k,n 2,k,n such that lim k (ζ ζ ) = + and such that for all suﬃciently large k k 2,k,n 1,k,n Γ (ζ) Γ (ζ) ≥ inf min , e3D . ζ∈(ζ ,ζ ) Γ (0) Γ (1) 0 1,k,n 2,k,n 33 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Combining

text: the

text: above

text: with

text: we

text: obtain

text: that

text: w.h.p.,

text: 2Λ

text: σ2

text: p

text: σ2

text: σ2

text: or

text: w.h.p.,

text: 2Λ

text: p

text: σ2

text: σ2

text: from

text: which

text: we

text: obtain

text: a

text: simpler

text: bound

text: σ2

text: namely

text: Λ

text: log

text: c

text: −p

text: log

text: k

text: σ2

text: or

text: log

text: c

text: Λ

text: p

text: log

text: k

text: σ2

text: Since

text: by

text: the

text: assumption

text: of

text: the

text: theorem

text: we

text: have

text: we

text: obtain

text: that

text: Λp

text: w.h.p.

text: as

text: k

text: k

text: Now

text: we

text: reapply

text: this

text: bound

text: for

text: and

text: obtain

text: that

text: w.h.p.

text: σ2

text: σ2

text: Taking

text: logarithm

text: of

text: both

text: sides,

text: we

text: obtain

text: that

text: w.h.p.

text: Λ

text: log

text: log

text: log

text: c

text: p

text: σ2

text: σ2

text: k

text: Now

text: again

text: since

text: it

text: is

text: easy

text: to

text: see

text: that

text: the

text: ratio

text: of

text: two

text: logarithms

text: approaches

text: unity

text: as

text: k

text: increases,

text: and

text: thus

text: the

text: limit

text: of

text: the

text: left-hand

text: side

text: is

text: in

text: the

text: limit.

text: Thus

text: Λ

text: /k

text: p

text: approaches

text: unity

text: in

text: the

text: limit

text: w.h.p.

text: as

text: k

text: increases.

text: This

text: completes

text: the

text: proof.

text: The

text: Overlap

text: Gap

text: Property

text: In

text: this

text: section

text: we

text: prove

text: Theorem

text: We

text: begin

text: by

text: establishing

text: a

text: certain

text: property

text: regarding

text: the

text: the

text: limiting

text: curve

text: function

text: Γ.

text: Lemma

text: Under

text: the

text: assumption

text: of

text: sequences

text: ζ

text: ζ

text: such

text: that

text: lim

text: k

text: (ζ

text: ζ

text: and

text: such

text: that

text: for

text: all

text: suﬃciently

text: large

text: k

text: k

text: Γ

text: (ζ)

text: Γ

text: (ζ)

text: inf

text: min

text: ζ∈(ζ

text: ,ζ

text: Γ

text: Γ

text: ##LTLine##

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text: (cid:0) (cid:1) √ − Proof. Recall that Γ (0) = σ and Γ (1) = 2k + σ2 exp k log p . We will rely on the results of ∗ n Proposition 2.4 and thus recall the deﬁnition of n . The proof proceeds by considering two cases. Assume ﬁrst (e7D2 + 1) σ2 log p ≤ n ≤ n∗. We choose ζ = e7D02σ2 , ζ = e7D02σ2 . Since → 0 ∞ 1,k,n 2k 2,k,n k by assumption σ2 + , then − 1 → ∞ k (ζ ζ ) = e7D2σ2 + . 2,k,n 1,k,n 2 0 ∗ ≥ Now since n < n , and therefore by Proposition 2.4, Γ (0) > Γ (1), it suﬃces to show Γ(ζ) e3D ∈ (cid:18) (cid:19) Γ(0) 0 for all ζ (ζ , ζ ). Since Γ is log-concave, it is suﬃcient to show that 1,k,n 2,k,n Γ (ζ ) Γ (ζ ) ≥ min 1,k,n , 2,k,n e3D . 0 Γ (0) Γ (0) Plugg(cid:32)ing in the values Γ (ζ ) , Γ ((cid:33)ζ ), we have (cid:18) (cid:19) (cid:18) 1,k,n (cid:19) 2,k,n (cid:18) (cid:19) (cid:0) (cid:1) (cid:0) (cid:1) min Γ (ζ1,k,n) 2 , Γ (ζ2,k,n) 2 = m(cid:0) in e7D(cid:1)2 + 1 e−e7D02σ2 logp , 2e7D2 + 1 e−e7D02σ2 logp 2n n Γ (0) Γ (0) 0 0 (cid:0) (cid:1) ≥ e7D2 + 1 e−e7D02σ2 logp n 0 ≥ − e7D2 + 1 e 1 ≥ 0 e6D2, 0 ≥ using our assumption that n e7D2σ2 log p. ∗ ≤ 0 Assume now n n < k log p . In this case we choose ζ = 1 and ζ = 1. Then − → ∞ 3(cid:32)log(cid:18)D≥0 ∗ (cid:19) (cid:18) (cid:19) (cid:33) 1,k,n 5 2,k,n 4 k (ζ ζ ) + . Since n n it suﬃces to show 2,k,n 1,k,n Γ (ζ ) 2 Γ (ζ ) 2 min 1,k,n , 2,k,n > e6D2. Γ (1) Γ (1) 0 (cid:32) (cid:33) (cid:32) (cid:33) (cid:18) (cid:19) (cid:18) (cid:19) But since n < k log p/(3 log D ) have 0 Γ (ζ ) 2 Γ (ζ ) 2 (cid:18) 2k + σ2 (cid:19) 3k + σ2 min 1,k,n , 2,k,n = min 5 e4klogp , 4 e3klogp 5n 4n Γ (1) Γ (1) 2k + σ2 2k + σ2 ≥ 1 12 2 9 min D 5 , D 4 4 0 3 0 > e6D2, 0 for all suﬃciently large D . This completes the proof of the lemma. 0 Now we return to the proof of Theorem 2.5. (cid:48) (cid:48) Proof of Theorem 2.5. Choose 0 < ζ < ζ < 1 from Lemma 6.1 and set r = D max (Γ (0) , Γ (1)). 1,k,n 2,k,n − (cid:48) −k (cid:48) 0 We will now prove that for this value of r and ζ = 1 ζ , ζ = 1 ζ , the set S k 1,k,n 2,k,n 2,k,n 1,k,n ∗ ∈ rk satisﬁes the claim of the theorem. Applying the second part of Theorem 2.1 we obtain β S r k 34 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Proof.

text: Recall

text: that

text: Γ

text: σ

text: and

text: Γ

text: σ2

text: exp

text: k

text: log

text: p

text: We

text: will

text: rely

text: on

text: the

text: results

text: of

text: n

text: Proposition

text: and

text: thus

text: recall

text: the

text: deﬁnition

text: of

text: n

text: The

text: proof

text: proceeds

text: by

text: considering

text: two

text: cases.

text: Assume

text: ﬁrst

text: σ2

text: log

text: p

text: n

text: n∗.

text: We

text: choose

text: ζ

text: ζ

text: Since

text: k

text: by

text: assumption

text: σ2

text: then

text: k

text: (ζ

text: ζ

text: Now

text: since

text: n

text: n

text: and

text: therefore

text: by

text: Proposition

text: Γ

text: Γ

text: it

text: suﬃces

text: to

text: show

text: Γ(ζ)

text: for

text: all

text: ζ

text: (ζ

text: ζ

text: Since

text: Γ

text: is

text: log-concave,

text: it

text: is

text: suﬃcient

text: to

text: show

text: that

text: Γ

text: (ζ

text: Γ

text: (ζ

text: min

text: Γ

text: Γ

text: in

text: the

text: values

text: Γ

text: (ζ

text: Γ

text: we

text: have

text: min

text: Γ

text: Γ

text: in

text: logp

text: logp

text: n

text: Γ

text: Γ

text: logp

text: n

text: e

text: using

text: our

text: assumption

text: that

text: n

text: log

text: p.

text: Assume

text: now

text: n

text: n

text: k

text: log

text: p

text: In

text: this

text: case

text: we

text: choose

text: ζ

text: and

text: ζ

text: Then

text: k

text: (ζ

text: ζ

text: Since

text: n

text: n

text: it

text: suﬃces

text: to

text: show

text: Γ

text: (ζ

text: Γ

text: (ζ

text: min

text: Γ

text: Γ

text: But

text: since

text: n

text: k

text: log

text: log

text: D

text: have

text: Γ

text: (ζ

text: Γ

text: (ζ

text: σ2

text: σ2

text: min

text: min

text: Γ

text: Γ

text: σ2

text: σ2

text: min

text: D

text: D

text: for

text: all

text: suﬃciently

text: large

text: D

text: This

text: completes

text: the

text: proof

text: of

text: the

text: lemma.

text: Now

text: we

text: return

text: to

text: the

text: proof

text: of

text: Theorem

text: Proof

text: of

text: Theorem

text: Choose

text: ζ

text: ζ

text: from

text: Lemma

text: and

text: set

text: r

text: D

text: max

text: (Γ

text: Γ

text: −k

text: We

text: will

text: now

text: prove

text: that

text: for

text: this

text: value

text: of

text: r

text: and

text: ζ

text: ζ

text: ζ

text: ζ

text: the

text: set

text: S

text: k

text: rk

text: satisﬁes

text: the

text: claim

text: of

text: the

text: theorem.

text: Applying

text: the

text: second

text: part

text: of

text: Theorem

text: we

text: obtain

text: β

text: S

text: r

text: k

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [FR13] Simon Foucart and Holger Rauhut, A mathematical introduction to compressive sensing, Springer, 2013. [GL16] David Gamarnik and Quan Li, Finding a large submatrix of a gaussian random matrix, arXiv preprint arXiv:1602.08529 (2016). [GSa] David Gamarnik and Madhu Sudan, Limits of local algorithms over sparse random graphs, Annals of Probability. To appear. [GSb] , Performance of sequential local algorithms for the random nae-k-sat prob- lem, SIAM Journal on Computing. To appear. [MB06] Nicolai Meinshausen and Peter Bu¨hlmann, High-dimensional graphs and variable selection with the lasso, Ann. Statist. 34 (2006), no. 3, 1436–1462. [MRT11] Andrea Montanari, Ricardo Restrepo, and Prasad Tetali, Reconstruction and clus- tering in random constraint satisfaction problems, SIAM Journal on Discrete Math- ematics 25 (2011), no. 2, 771–808. [RV14] Mustazee Rahman and Balint Virag, Local algorithms for independent sets are half- optimal, arXiv preprint arXiv:1402.0485 (2014). [SW95] A. Shwartz and A. Weiss, Large deviations for performance analysis, Chapman and Hall, 1995. [Wai09a] Martin J Wainwright, Information-theoretic limits on sparsity recovery in the high- dimensional and noisy setting, Information Theory, IEEE Transactions on 55 (2009), no. 12, 5728–5741. [Wai09b] , Sharp thresholds for high-dimensional and noisy sparsity recovery using constrained quadratic programming (lasso), IEEE transactions on information the- ory 55 (2009), no. 5, 2183–2202. [WWR10] Wei Wang, Martin J Wainwright, and Kannan Ramchandran, Information-theoretic limits on sparse signal recovery: Dense versus sparse measurement matrices, Infor- mation Theory, IEEE Transactions on 56 (2010), no. 6, 2967–2979. \ [ZMWL15] Le Zheng, Arian Maleki, Xiaodong Wang, and Teng Long, Does ell p-minimization \ outperform ell 1-minimization?, arXiv preprint arXiv:1501.03704 (2015). [ZY06] Peng Zhao and Bin Yu, On model selection consistency of lasso, J. Mach. Learn. Res. 7 (2006), 2541–2563. 37 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: reference

text: Simon

text: Foucart

text: and

text: Holger

text: Rauhut,

text: A

text: mathematical

text: introduction

text: to

text: compressive

text: sensing,

text: Springer,

text: David

text: Gamarnik

text: and

text: Quan

text: Li,

text: Finding

text: a

text: large

text: submatrix

text: of

text: a

text: gaussian

text: random

text: matrix,

text: arXiv

text: preprint

text: [GSa]

text: David

text: Gamarnik

text: and

text: Madhu

text: Sudan,

text: Limits

text: of

text: local

text: algorithms

text: over

text: sparse

text: random

text: graphs,

text: Annals

text: of

text: Probability.

text: To

text: appear.

text: [GSb]

text: Performance

text: of

text: sequential

text: local

text: algorithms

text: for

text: the

text: random

text: nae-k-sat

text: prob-

text: lem,

text: SIAM

text: Journal

text: on

text: Computing.

text: To

text: appear.

text: Nicolai

text: Meinshausen

text: and

text: Peter

text: Bu¨hlmann,

text: High-dimensional

text: graphs

text: and

text: variable

text: selection

text: with

text: the

text: lasso,

text: Ann.

text: Statist.

text: no.

text: Andrea

text: Montanari,

text: Ricardo

text: Restrepo,

text: and

text: Prasad

text: Tetali,

text: Reconstruction

text: and

text: clus-

text: tering

text: in

text: random

text: constraint

text: satisfaction

text: problems,

text: SIAM

text: Journal

text: on

text: Discrete

text: Math-

text: ematics

text: no.

text: Mustazee

text: Rahman

text: and

text: Balint

text: Virag,

text: Local

text: algorithms

text: for

text: independent

text: sets

text: are

text: half-

text: optimal,

text: arXiv

text: preprint

text: A.

text: Shwartz

text: and

text: A.

text: Weiss,

text: Large

text: deviations

text: for

text: performance

text: analysis,

text: Chapman

text: and

text: Hall,

text: Martin

text: J

text: Wainwright,

text: Information-theoretic

text: limits

text: on

text: sparsity

text: recovery

text: in

text: the

text: high-

text: dimensional

text: and

text: noisy

text: setting,

text: Information

text: Theory,

text: IEEE

text: Transactions

text: on

text: no.

text: Sharp

text: thresholds

text: for

text: high-dimensional

text: and

text: noisy

text: sparsity

text: recovery

text: using

text: constrained

text: quadratic

text: programming

text: (lasso),

text: IEEE

text: transactions

text: on

text: information

text: the-

text: ory

text: no.

text: Wei

text: Wang,

text: Martin

text: J

text: Wainwright,

text: and

text: Kannan

text: Ramchandran,

text: Information-theoretic

text: limits

text: on

text: sparse

text: signal

text: recovery:

text: Dense

text: versus

text: sparse

text: measurement

text: matrices,

text: Infor-

text: mation

text: Theory,

text: IEEE

text: Transactions

text: on

text: no.

text: Le

text: Zheng,

text: Arian

text: Maleki,

text: Xiaodong

text: Wang,

text: and

text: Teng

text: Long,

text: Does

text: ell

text: p-minimization

text: outperform

text: ell

text: arXiv

text: preprint

text: Peng

text: Zhao

text: and

text: Bin

text: Yu,

text: On

text: model

text: selection

text: consistency

text: of

text: lasso,

text: J.

text: Mach.

text: Learn.

text: Res.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 11 Channel interpolation arises when a channel behaves as W with probability α and as W with probability − 1 2 1 α. The transmitter has no control on which behavior the channel chooses, but on the other hand, the receiver knows which one was chosen. Channel interpolations were used in [12] to construct interpolations between polar codes and Reed-Muller codes. ∗ X (cid:88) ∈ − ∈ ∈ Now ﬁx a binary operation on . For every W DMCX Y, deﬁne W DMCX Y and W + , , 2 DMCX Y ×X as: , 2 − | 1 | ∗ | W (y , y u ) = |X | W (y u u )W (y u ), 1 2 1 1 1 2 2 2 ∈X u2 and | 1 | ∗ | W +(y , y , u u ) = |X |W (y u u )W (y u ). 1 2 1 2 1 1 2 2 2 These operations generalize Arıkan’s polarization transformations [13]. Proposition 3. We have: → ◦ × • The mapping (W, V ) V W from DMCX Y DMCY Z to DMCX Z is c(cid:96)ontinuo(cid:96)us. → ⊕ , × , , • The mapping (W , W ) W W from DMCX Y DMCX Y to DMCX X Y Y is continuous. • The mapping (W1 , W2 ) → W1 ⊗ W2 from DMCX1, Y1 × DMCX2, Y2 to DMC1X ×X2, Y1 ×Y2 is continu(cid:96)ous. 1 2 →1 2 − 1, 1 2, ×2 1× 2, 1 2 • The mapping (W , W , α) [αW , (1 α)W ] from DMCX Y DMCX Y [0, 1] to DMCX Y Y 1 2 1 2 , 1 , 2 , 1 2 is continuous. ∗ X → − • For any binary operation on , the mapping W W from DMCX Y to DMCX Y is continuous. ∗ X → , , 2 • For any binary operation on , the mapping W W + from DMCX Y to DMCX Y ×X is , , 2 continuous. Proof. The continuity immediately follows from the deﬁnitions. V. CONTINUITY ON DMC(Xo)Y , It is well known that the parameters deﬁned in section depend only on the RX(o)Y-equivalence class of ∈ , W . Therefore, we can deﬁne those parameters for any Wˆ DMC(Xo)Y through the transcendent mapping , (deﬁned in Lemma 2). The following proposition shows that those parameters are continuous on DMC(Xo)Y: , Proposition 4. We have: × → R • I : ∆X DMC(Xo)Y + is continuous and concave in p. → R, • C : DMC(Xo)Y + is continuous. × , → • P : ∆X DMC(Xo)Y [0, 1] is continuous and concave in p. e → , • Z : DMC(Xo)Y [0, 1] is continuous. , C X → • For every code on , P C : DMC(Xo)Y [0, 1] is continuous. e, ≤ ≤ |,X | → • For every n > 0 and every 1 M n, the mapping P : DMC(Xo)Y [0, 1] is continuous. e,n,M , Proof. Since the corresponding parameters are continuous on DMCX Y (Proposition 2), Lemma 2 implies , that they are continuous on DMC(Xo)Y. The only cases that need a special treatment are those of I and Z. , We will only prove the continuity of I since the proof of continuity of Z is similar. × Deﬁne the relation R on ∆X DMCX Y as , ⇔ (p , W )R(p , W ) p = p and W RX(o)YW . 1 1 2 2 1 2 1 , 2 It is easy to see that I(p, W ) depends only on the R-equivalence class of (p, W ). Since I is continuous on × × ∆X DMCX Y, Lemma 2 implies that the transcendent mapping of I is continuous on (∆X DMCX Y)/R. , × , On the other hand, since ∆X is locally compact, Theorem 1 implies that (∆X DMCX Y)/R can be × × , identiﬁed with ∆X (DMCX Y /RX(o)Y) = ∆X DMC(Xo)Y and the two spaces have the same topology. , × , , Therefore, I is continuous on ∆X DMC(Xo)Y. , ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Channel

text: interpolation

text: arises

text: when

text: a

text: channel

text: behaves

text: as

text: W

text: with

text: probability

text: α

text: and

text: as

text: W

text: with

text: probability

text: α.

text: The

text: transmitter

text: has

text: no

text: control

text: on

text: which

text: behavior

text: the

text: channel

text: chooses,

text: but

text: on

text: the

text: other

text: hand,

text: the

text: receiver

text: knows

text: which

text: one

text: was

text: chosen.

text: Channel

text: interpolations

text: were

text: used

text: in

text: to

text: construct

text: interpolations

text: between

text: polar

text: codes

text: and

text: Reed-Muller

text: codes.

text: X

text: Now

text: ﬁx

text: a

text: binary

text: operation

text: on

text: For

text: every

text: W

text: DMCX

text: Y,

text: deﬁne

text: W

text: DMCX

text: Y

text: and

text: W

text: DMCX

text: Y

text: ×X

text: as:

text: W

text: (y

text: y

text: u

text: |X

text: W

text: (y

text: u

text: u

text: )W

text: (y

text: u

text: ∈X

text: and

text: W

text: +(y

text: y

text: u

text: u

text: |X

text: |W

text: (y

text: u

text: u

text: )W

text: (y

text: u

text: These

text: operations

text: generalize

text: Arıkan’s

text: polarization

text: transformations

text: Proposition

text: We

text: have:

text: The

text: mapping

text: (W,

text: V

text: V

text: W

text: from

text: DMCX

text: Y

text: DMCY

text: Z

text: to

text: DMCX

text: Z

text: is

text: The

text: mapping

text: (W

text: W

text: W

text: W

text: from

text: DMCX

text: Y

text: DMCX

text: Y

text: to

text: DMCX

text: X

text: Y

text: Y

text: is

text: continuous.

text: The

text: mapping

text: from

text: to

text: is

text: →1

text: ×2

text: 1×

text: The

text: mapping

text: (W

text: W

text: α)

text: [αW

text: α)W

text: from

text: DMCX

text: Y

text: DMCX

text: Y

text: to

text: DMCX

text: Y

text: Y

text: is

text: continuous.

text: X

text: For

text: any

text: binary

text: operation

text: on

text: the

text: mapping

text: W

text: W

text: from

text: DMCX

text: Y

text: to

text: DMCX

text: Y

text: is

text: continuous.

text: X

text: For

text: any

text: binary

text: operation

text: on

text: the

text: mapping

text: W

text: W

text: from

text: DMCX

text: Y

text: to

text: DMCX

text: Y

text: ×X

text: is

text: continuous.

text: Proof.

text: The

text: continuity

text: immediately

text: follows

text: from

text: the

text: deﬁnitions.

text: V.

text: CONTINUITY

text: ON

text: DMC(Xo)Y

text: It

text: is

text: well

text: known

text: that

text: the

text: parameters

text: deﬁned

text: in

text: section

text: depend

text: only

text: on

text: the

text: RX(o)Y-equivalence

text: class

text: of

text: W

text: Therefore,

text: we

text: can

text: deﬁne

text: those

text: parameters

text: for

text: any

text: Wˆ

text: DMC(Xo)Y

text: through

text: the

text: transcendent

text: mapping

text: (deﬁned

text: in

text: Lemma

text: The

text: following

text: proposition

text: shows

text: that

text: those

text: parameters

text: are

text: continuous

text: on

text: DMC(Xo)Y:

text: Proposition

text: We

text: have:

text: R

text: I

text: ∆X

text: DMC(Xo)Y

text: is

text: continuous

text: and

text: concave

text: in

text: p.

text: R,

text: C

text: DMC(Xo)Y

text: is

text: continuous.

text: P

text: ∆X

text: DMC(Xo)Y

text: is

text: continuous

text: and

text: concave

text: in

text: p.

text: e

text: Z

text: DMC(Xo)Y

text: is

text: continuous.

text: C

text: X

text: For

text: every

text: code

text: on

text: P

text: C

text: DMC(Xo)Y

text: is

text: continuous.

text: e,

text: |,X

text: For

text: every

text: n

text: and

text: every

text: M

text: n,

text: the

text: mapping

text: P

text: DMC(Xo)Y

text: is

text: continuous.

text: e,n,M

text: Proof.

text: Since

text: the

text: corresponding

text: parameters

text: are

text: continuous

text: on

text: DMCX

text: Y

text: (Proposition

text: Lemma

text: implies

text: that

text: they

text: are

text: continuous

text: on

text: DMC(Xo)Y.

text: The

text: only

text: cases

text: that

text: need

text: a

text: special

text: treatment

text: are

text: those

text: of

text: I

text: and

text: Z.

text: We

text: will

text: only

text: prove

text: the

text: continuity

text: of

text: I

text: since

text: the

text: proof

text: of

text: continuity

text: of

text: Z

text: is

text: similar.

text: Deﬁne

text: the

text: relation

text: R

text: on

text: ∆X

text: DMCX

text: Y

text: as

text: (p

text: W

text: )R(p

text: W

text: p

text: p

text: and

text: W

text: RX(o)YW

text: It

text: is

text: easy

text: to

text: see

text: that

text: I(p,

text: W

text: depends

text: only

text: on

text: the

text: R-equivalence

text: class

text: of

text: (p,

text: W

text: Since

text: I

text: is

text: continuous

text: on

text: ∆X

text: DMCX

text: Y,

text: Lemma

text: implies

text: that

text: the

text: transcendent

text: mapping

text: of

text: I

text: is

text: continuous

text: on

text: (∆X

text: DMCX

text: Y)/R.

text: On

text: the

text: other

text: hand,

text: since

text: ∆X

text: is

text: locally

text: compact,

text: Theorem

text: implies

text: that

text: (∆X

text: DMCX

text: Y)/R

text: can

text: be

text: identiﬁed

text: with

text: ∆X

text: (DMCX

text: Y

text: /RX(o)Y)

text: ∆X

text: DMC(Xo)Y

text: and

text: the

text: two

text: spaces

text: have

text: the

text: same

text: topology.

text: Therefore,

text: I

text: is

text: continuous

text: on

text: ∆X

text: DMC(Xo)Y.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 13 V → Lemma 6. Let (S, ) be an arbitrary topological space. A mapping f : DMC(Xo)∗ S is continuous on T T , ≥ (DMC(Xo)∗, (Xo) ∗) if and only if it is continuous on (DMC(Xo) , X(o) ) for every n 1. , s, , ,[n] ,[n] Proof. T ⇔ − ∈ T ∀ ∈ V f is continuous on (DMC(Xo)∗, (Xo) ∗) f 1(V ) (Xo) ∗ V , s, , s, , ⇔ − ∩ ∈ T ∀ ≥ ∀ ∈ V f 1(V ) DMC(Xo) X(o) n 1, V ,[n] ,[n] ⇔ T ∀ ≥ f is continuous on (DMC(Xo) , X(o) ) n 1. ,[n] ,[n] (cid:91) ≥ Since the channel parameters I, C, P , Z, P C and P are deﬁned on DMC(Xo) for every n 1 e e, e,n,M ,[n] (see Section ), they are also deﬁned on DMC(Xo)∗ = DMC(Xo) . The following proposition shows that , ,[n] ≥ n 1 those parameters are continuous in the strong topology: U Proposition 6. Let X be the standard topology on ∆X . We have: × → R × U ⊗ T • I : ∆X DMC(Xo)∗ + is continuous on (∆X DMC(Xo)∗, X (Xo) ∗) and concave in p. → R, T , s, , • C : DMC(Xo)∗ + is continuous on (DMC(Xo)∗, (Xo) ∗). × , → , s, ×, U ⊗ T • P : ∆X DMC(Xo)∗ [0, 1] is continuous on (∆X DMC(Xo)∗, X (Xo) ∗) and concave in p. e → , T , s, , • Z : DMC(Xo)∗ [0, 1] is continuous on (DMC(Xo)∗, (Xo) ∗). , C X → , s, , T • For every code on , P C : DMC(Xo)∗ [0, 1] is continuous on (DMC(Xo)∗, (Xo) ∗). e, ≤ ≤ |,X | ,→ s, , • For every n > 0 and every 1 M n, the mapping P : DMC(Xo)∗ [0, 1] is continuous on T e,n,M , (DMC(Xo)∗, (Xo) ∗). , s, , Proof. The continuity of C, Z, P and P immediately follows from Proposition 4 and Lemma 6. e,C e,n,M Since the proofs of continuity of I and Z are sim(cid:97)ilar,(cid:0)we only prove th(cid:1)e continuity for I. Due to the distributivity of the product with respect to disjoint unions, we have × × ∆X DMCX ∗ = ∆X DMCX , , ,[n] ≥ n(cid:77)1 (cid:0) (cid:1) and U ⊗ T U ⊗ T X X ∗ = X X . s, , ,[n] ≥ n 1 × U ⊗T × Therefore, (∆X DMCX ∗, X X ∗) is the disjoint union of the spaces (∆X DMCX ) ≥ . Moreover, ,× s, , ≥ ,[n] n 1 × I is continuous on ∆X DMCX for every n 1. We conclude that I is continuous on (∆X U ⊗ T ,[n] DMCX ∗, X X ∗). , s, , × (cid:0) (cid:1) Deﬁne the relation R on ∆X DMCX ∗ as follows: (p , W )R(p , W ) if and only if p = p and , 1 1 2 2 1 (cid:0) 2 W RX(o)∗W . Since I(p, W ) depends only on the R-equivalence class of (p, W ), Lemma 2 shows that 1 , 2 (cid:1) (cid:0) × U ⊗ T (cid:1) the transcendent map of I is a continuous mapping from (∆X DMCX ∗)/R, ( X X ∗)/R to R , s, , × +. On the other hand, since ∆X is locally compact and Hausdorff, Theorem 1 implies that (∆X U ⊗ T × U ⊗ T DMCX ∗)/R, ( X X ∗)/R can be identiﬁed with ∆X (DMCX ∗ /RX(o)∗), X ( X ∗/RX(o)∗) = ×, U ⊗s,T , × , , U ⊗ T s, , , (∆X DMC(Xo)∗, X (Xo) ∗). Therefore, I is continuous on (∆X DMC(Xo)∗, X (Xo) ∗). , s, , , s, , It is also possible to extend the deﬁnition of all the channel operations that were deﬁned in section to DMC(Xo)∗. Moreover, it is possible to show that many channel operations are continuous in the strong , topology: ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: V

text: Lemma

text: Let

text: (S,

text: be

text: an

text: arbitrary

text: topological

text: space.

text: A

text: mapping

text: f

text: DMC(Xo)∗

text: S

text: is

text: continuous

text: on

text: T

text: T

text: (DMC(Xo)∗,

text: (Xo)

text: if

text: and

text: only

text: if

text: it

text: is

text: continuous

text: on

text: (DMC(Xo)

text: X(o)

text: for

text: every

text: n

text: s,

text: ,[n]

text: ,[n]

text: Proof.

text: T

text: T

text: V

text: f

text: is

text: continuous

text: on

text: (DMC(Xo)∗,

text: (Xo)

text: f

text: (Xo)

text: V

text: s,

text: s,

text: T

text: V

text: f

text: DMC(Xo)

text: X(o)

text: n

text: V

text: ,[n]

text: ,[n]

text: T

text: f

text: is

text: continuous

text: on

text: (DMC(Xo)

text: X(o)

text: n

text: ,[n]

text: ,[n]

text: Since

text: the

text: channel

text: parameters

text: I,

text: C,

text: P

text: Z,

text: P

text: C

text: and

text: P

text: are

text: deﬁned

text: on

text: DMC(Xo)

text: for

text: every

text: n

text: e

text: e,

text: e,n,M

text: ,[n]

text: (see

text: Section

text: they

text: are

text: also

text: deﬁned

text: on

text: DMC(Xo)∗

text: DMC(Xo)

text: The

text: following

text: proposition

text: shows

text: that

text: ,[n]

text: n

text: those

text: parameters

text: are

text: continuous

text: in

text: the

text: strong

text: topology:

text: U

text: Proposition

text: Let

text: X

text: be

text: the

text: standard

text: topology

text: on

text: ∆X

text: We

text: have:

text: R

text: U

text: T

text: I

text: ∆X

text: DMC(Xo)∗

text: is

text: continuous

text: on

text: (∆X

text: DMC(Xo)∗,

text: X

text: (Xo)

text: and

text: concave

text: in

text: p.

text: R,

text: T

text: s,

text: C

text: DMC(Xo)∗

text: is

text: continuous

text: on

text: (DMC(Xo)∗,

text: (Xo)

text: s,

text: U

text: T

text: P

text: ∆X

text: DMC(Xo)∗

text: is

text: continuous

text: on

text: (∆X

text: DMC(Xo)∗,

text: X

text: (Xo)

text: and

text: concave

text: in

text: p.

text: e

text: T

text: s,

text: Z

text: DMC(Xo)∗

text: is

text: continuous

text: on

text: (DMC(Xo)∗,

text: (Xo)

text: C

text: X

text: s,

text: T

text: For

text: every

text: code

text: on

text: P

text: C

text: DMC(Xo)∗

text: is

text: continuous

text: on

text: (DMC(Xo)∗,

text: (Xo)

text: e,

text: |,X

text: s,

text: For

text: every

text: n

text: and

text: every

text: M

text: n,

text: the

text: mapping

text: P

text: DMC(Xo)∗

text: is

text: continuous

text: on

text: T

text: e,n,M

text: (DMC(Xo)∗,

text: (Xo)

text: s,

text: Proof.

text: The

text: continuity

text: of

text: C,

text: Z,

text: P

text: and

text: P

text: immediately

text: follows

text: from

text: Proposition

text: and

text: Lemma

text: e,C

text: e,n,M

text: Since

text: the

text: proofs

text: of

text: continuity

text: of

text: I

text: and

text: Z

text: are

text: only

text: prove

text: continuity

text: for

text: I.

text: Due

text: to

text: the

text: distributivity

text: of

text: the

text: product

text: with

text: respect

text: to

text: disjoint

text: unions,

text: we

text: have

text: ∆X

text: DMCX

text: ∆X

text: DMCX

text: ,[n]

text: and

text: U

text: T

text: U

text: T

text: X

text: X

text: X

text: X

text: s,

text: ,[n]

text: n

text: U

text: ⊗T

text: Therefore,

text: (∆X

text: DMCX

text: X

text: X

text: is

text: the

text: disjoint

text: union

text: of

text: the

text: spaces

text: (∆X

text: DMCX

text: Moreover,

text: s,

text: ,[n]

text: n

text: I

text: is

text: continuous

text: on

text: ∆X

text: DMCX

text: for

text: every

text: n

text: We

text: conclude

text: that

text: I

text: is

text: continuous

text: on

text: (∆X

text: U

text: T

text: ,[n]

text: DMCX

text: X

text: X

text: s,

text: Deﬁne

text: the

text: relation

text: R

text: on

text: ∆X

text: DMCX

text: as

text: follows:

text: (p

text: W

text: )R(p

text: W

text: if

text: and

text: only

text: if

text: p

text: p

text: and

text: W

text: RX(o)∗W

text: Since

text: I(p,

text: W

text: depends

text: only

text: on

text: the

text: R-equivalence

text: class

text: of

text: (p,

text: W

text: Lemma

text: shows

text: that

text: U

text: T

text: the

text: transcendent

text: map

text: of

text: I

text: is

text: a

text: continuous

text: mapping

text: from

text: (∆X

text: DMCX

text: ∗)/R,

text: X

text: X

text: ∗)/R

text: to

text: R

text: s,

text: On

text: the

text: other

text: hand,

text: since

text: ∆X

text: is

text: locally

text: compact

text: and

text: Hausdorff,

text: Theorem

text: implies

text: that

text: (∆X

text: U

text: T

text: U

text: T

text: DMCX

text: ∗)/R,

text: X

text: X

text: ∗)/R

text: can

text: be

text: identiﬁed

text: with

text: ∆X

text: (DMCX

text: /RX(o)∗),

text: X

text: X

text: ∗/RX(o)∗)

text: U

text: ⊗s,T

text: U

text: T

text: s,

text: (∆X

text: DMC(Xo)∗,

text: X

text: (Xo)

text: Therefore,

text: I

text: is

text: continuous

text: on

text: (∆X

text: DMC(Xo)∗,

text: X

text: (Xo)

text: s,

text: s,

text: It

text: is

text: also

text: possible

text: to

text: extend

text: the

text: deﬁnition

text: of

text: all

text: the

text: channel

text: operations

text: that

text: were

text: deﬁned

text: in

text: section

text: to

text: DMC(Xo)∗.

text: Moreover,

text: it

text: is

text: possible

text: to

text: show

text: that

text: many

text: channel

text: operations

text: are

text: continuous

text: in

text: the

text: strong

text: topology:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##