text: 7 a given network [8]. Finally, parameters η and η Algorithm 1: KroMFac select detect are arbitrarily small positive constants used as stopping Input: G, M, C, N , Θ , α, β, (cid:15), δ, η , η , λ 0 init select detect criteria during node selection and community detection (i.e., Output: ψ D ← ∞ ← {∅} convergence criteria). As the output of Algorithm 1, we 1 Ini∈tia{liza·t·i·on: } min ; ψ[c] for c 1, , C deﬁne ψ as the set of detected communities. 2 functio←n KroMFac We assume that all communities initially have no mem- 3 A Adjacency matrix of G ← bers. To ﬁnd the true community structures of the incom- 4 A(RM) ←GraphRecv(A, N0, Θinit, M) plete input graph, we ﬁrst fully recover the graph (refer to 5 (H, r) NodeSelect(A(RM), M, α, β, (cid:15), ηselect) Algorithm 2). By analyzing the recovered graph, we then 6 for i from 1 to H do ← { · · · } select H inﬂuential nodes and determine the ranking vector 7 R(i) Connect nodes r[1], , r[i] to G ∈ ← r NH that represents the indices of ranked inﬂuential 8 A(i) Adjacency matrix of R(i) DR ← nodes and plays a crucial role in community detection 9 (D ←, FD(i))− CommunDet(A(Ri), C, ηdetect) accuracy (refer to Algorithm 3). In this step, speciﬁcally, 10 D λDlog(i + 1) we solve the joint optimization problem described in (5) 11 if mi←n < then 12 Fˆ F(i) through exhaustive search over i by sequentially connecting ← inﬂuential nodes to the existing observable graph based on 13 ˆiD i ← D 14 min the order in r. For each i, we acquire a corresponded graph R(i) and its adjacency matrix A(i) (see Deﬁnition 1). By 15 for u from 1 to (N + M) do using Algorithm 4, we are then Rcapable of obtaining the 16 for c from≥1 to C do D 17 if Fuc ←δ then ∪ { } loss function associated with afﬁliation matDrix F given 18 ψ[c] ψ[c] u the input matrix A(i). As a result, we obtain as the DR − min smallest value of λ log(i + 1) (i.e., the regularized 19 return ψ loss), which in turn provides us with the optimal Fˆ and ˆi. Every entry Fˆ in the optimal afﬁliation matrix Fˆ then uc ∈ ∪ Algorithm 2: GraphRecv describes the likelihood that the node u (V V ) belongs ∈ { · · · } M Input: A, N , Θ , M to community c 1, , C . Therefore, it is possible to 0 init Output: A(M) recover the community structures ψ by assigning node u to R ← 1 Initialization: K 0 community c if the corresponding entry Fˆuc is greater than 2 function GraphRecv or equal to the threshold δ. In the following, we elaborate 3 while ←N0K < (N + M) do on each major phase of the KroMFac framework. 4 K K + 1 ← 4.1.1 Graph Recovery 5 (Θ, σ←) EM(A, N0, Θinit, K) 6 ΘK Θ − The ﬁrst step of our framework is the inference of 7 for u fro←m 1 to (K 1) do missing parts of the graph from priors on the observable 8 Θu ←ΘK⊗ matrix A and the number of missing nodes, M, by using 9 ΘK Θ Θu the function GraphRecv in Algorithm 2. By applying the 10 for u from 1 to (N + M) do EM algorithm, from an initialized Θinit, the E-step samples 11 for v fro≤m 1 to (N +≤M) do the missing parts Z1 and Z2, and the permutation σ. In the 12 if u N and←v N then(cid:16) (cid:17) M-step, a stochastic gradient descent process subsequently 13 [A(RM≤)]uv Auv 14 else if v u then optimizes the parameter matrix Θ given the samples ob- ∼ tained in the E-step. The EM iteration alternates between 15 [A(RM)]uv Bernoulli ΘKσ(u)σ(v) ← performing the E-step and M-step according to the follow- 16 [A(RM)]vu [A(RM)]uv ing expressions, respectively: (cid:16) (cid:17) 17 return A(M) R E-step: ∼ | Z(t), Z(t), σ(t) P(Z , Z , σ A, Θ(t)), 1 2 1 2 result, the EM function returns both Θ = Θ(t) and σ = σ(t) M-step: (see line 5 in Algorithm 2). We then generate the stochastic Θ(t+1) = arg max E[P(Z(t), Z(t), σ(t), A|Θ)], adjacency matrix ΘK. To create the fully recovered matrix Θ∈(0,1)N0 1 2 A(RM), for the ﬁrst N rows and columns of A(RM), we replicate the entries of matrix A in the upper left of matrix where the supercript (t) denotes the iteration index. As a A(M) (refer to lines 12–13 in the algorithm). To infer the R ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: 10 TABLE 2: LFR parameters of the ﬁve synthetic graphs Description Graph 1 Graph 2 Graph 3 Graph 4 Graph 5 the number of nodes 1,000 1,000 1,000 1,000 1,000 average degree 25 40 30 40 40 maximum degree 100 300 80 70 100 minimum community size 3 3 3 3 3 maximum community size 100 100 100 100 100 degree exponent 2 2 2 2.5 3 community size exponent 1.5 1 1.5 1 1 mixing parameter 0.1 0.3 0.2 0.2 0.1 the number of overlapping nodes 200 100 100 200 200 the number of communities per node 10 10 10 10 10 matrix F is found given an adjacency matrix A(M) that can degree distribution and overlapping communities. To create R be ﬁrst inferred by the graph recovery phase. The problem an LFR graph, ten parameters need to be speciﬁed, which formulation is thus given by are summarized in Table 2. While parameters such as the | number of nodes, average degree, maximum degree, min- Fˆ = arg max P(A(M) F), ≥ R imum community size, degree exponent, and community F 0 size exponent are rather straightforward to understand, we which corresponds to another special case with no regu- explain the remaining three parameters: larization term, where i is set to M in (1). Note that node • The mixing parameter, denoted by µ, controls the ranking is not necessary since all recovered nodes are in- proportion of random edges to total edges; for ex- serted into the graph. To solve this problem, we ﬁrst follow ample, if µ = 0.3, then the LFR benchmark produces the steps similar to those in Section to recover the matrix a graph such that approximately 70% of edges link A(M). After the graph recovery phase in Algorithm 2, the R to nodes within the same community, while the optimal F can be acquired via Algorithm 4 by assuming that remaining 30% connect to nodes in other randomly i = M. Then, the hard-decision process in Algorithm 1 is selected communities. In general, if µ is closer to one, performed to produce the ﬁnal result ψ. In the next section, then the community structures become weaker. we experimentally show that the above scheme that does • The number of overlapping nodes refers to the num- not employ the node ranking and selection phase (i.e., Base- ber of nodes in the graph that belong to more than line 2) signiﬁcantly degrades the detection performance, one community. compared to our original KroMFac framework. • The number of communities per node indicates the 5 EXPERIMENTAL EVALUATION average number of communities to which each of the In this section, we ﬁrst describe both synthetic and overlapping nodes belongs. real-world datasets. By adopting the NMI as a popular To cover various domains of network applications, we gen- information-theoretic performance metric, we then present erate ﬁve LFR graphs with differing parameter settings as the performance of our community detection framework speciﬁed in Table 2. and compare it against the two baseline schemes. 5.1 Datasets 5.1.2 Real-World Datasets To evaluate the community detection performance of To validate the applicability of our approach, two types our approach, we rely on datasets for which ground-truth of real-world datasets are also used for evaluation. More communities are explicitly labeled. In the following, both speciﬁcally, from the available SNAP datasets [39], we use synthetic and real-world datasets across various domains the Amazon product co-purchasing network [40] and the are taken into account. collaboration network of DBLP [41]. The statistics of these 5.1.1 Synthetic Datasets datasets are summarized in Table 3, and the basic character- We construct synthetic graphs via the extended istics of each network are described in the following: Lancichinetti-Fortunato-Radicchi (LFR) benchmark [38], • The number of nodes (N.N.): In the Amazon network, which is built upon a generative model that creates nodes nodes represent products. In DBLP, nodes represent along with prior known community labels. The benchmark authors. is capable of generating graphs that replicate important • The number of edges (N.E.): In the Amazon network, features of real social networks such as the power-law edges connect products that are commonly pur- ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: TABLE

text: LFR

label: caption

text: parameters

text: of

text: the

text: ﬁve

text: synthetic

text: graphs

text: Description

label: table

text: Graph

text: Graph

text: Graph

text: Graph

text: Graph

text: the

text: number

text: of

text: nodes

text: average

text: degree

text: maximum

text: degree

text: minimum

text: community

text: size

text: maximum

text: community

text: size

text: degree

text: exponent

text: community

text: size

text: exponent

text: mixing

text: parameter

text: the

text: number

text: of

text: overlapping

text: nodes

text: the

text: number

text: of

text: communities

text: per

text: node

text: matrix

text: F

text: is

text: found

text: given

text: an

text: adjacency

text: matrix

text: A(M)

text: that

text: can

text: degree

text: distribution

text: and

text: overlapping

text: communities.

text: To

text: create

text: R

text: be

text: ﬁrst

text: inferred

text: by

text: the

text: graph

text: recovery

text: phase.

text: The

text: problem

text: an

text: LFR

text: graph,

text: ten

text: parameters

text: need

text: to

text: be

text: speciﬁed,

text: which

text: formulation

text: is

text: thus

text: given

text: by

text: are

text: summarized

text: in

text: Table

text: While

text: parameters

text: such

text: as

text: the

text: number

text: of

text: nodes,

text: average

text: degree,

text: maximum

text: degree,

text: min-

text: Fˆ

text: arg

text: max

text: P(A(M)

text: F),

text: R

text: imum

text: community

text: size,

text: degree

text: exponent,

text: and

text: community

text: F

text: size

text: exponent

text: are

text: rather

text: straightforward

text: to

text: understand,

text: we

text: which

text: corresponds

text: to

text: another

text: special

text: case

text: with

text: no

text: regu-

text: explain

text: the

text: remaining

text: three

text: parameters:

text: larization

text: term,

text: where

text: i

text: is

text: set

text: to

text: M

text: in

text: Note

text: that

text: node

text: The

text: mixing

text: parameter,

text: denoted

text: by

text: µ,

text: controls

text: the

text: ranking

text: is

text: not

text: necessary

text: since

text: all

text: recovered

text: nodes

text: are

text: in-

text: proportion

text: of

text: random

text: edges

text: to

text: total

text: edges;

text: for

text: ex-

text: serted

text: into

text: the

text: graph.

text: To

text: solve

text: this

text: problem,

text: we

text: ﬁrst

text: follow

text: ample,

text: if

text: µ

text: then

text: the

text: LFR

text: benchmark

text: produces

text: the

text: steps

text: similar

text: to

text: those

text: in

text: Section

text: to

text: recover

text: the

text: matrix

text: a

text: graph

text: such

text: that

text: approximately

text: of

text: edges

text: link

text: A(M).

text: After

text: the

text: graph

text: recovery

text: phase

text: in

text: Algorithm

text: the

text: R

text: to

text: nodes

text: within

text: the

text: same

text: community,

text: while

text: the

text: optimal

text: F

text: can

text: be

text: acquired

text: via

text: Algorithm

text: by

text: assuming

text: that

text: remaining

text: connect

text: to

text: nodes

text: in

text: other

text: randomly

text: i

text: M.

text: Then,

text: the

text: hard-decision

text: process

text: in

text: Algorithm

text: is

text: selected

text: communities.

text: In

text: general,

text: if

text: µ

text: is

text: closer

text: to

text: one,

text: performed

text: to

text: produce

text: the

text: ﬁnal

text: result

text: ψ.

text: In

text: the

text: next

text: section,

text: then

text: the

text: community

text: structures

text: become

text: weaker.

text: we

text: experimentally

text: show

text: that

text: the

text: above

text: scheme

text: that

text: does

text: The

text: number

text: of

text: overlapping

text: nodes

text: refers

text: to

text: the

text: num-

text: not

text: employ

text: the

text: node

text: ranking

text: and

text: selection

text: phase

text: (i.e.,

text: Base-

text: ber

text: of

text: nodes

text: in

text: the

text: graph

text: that

text: belong

text: to

text: more

text: than

text: line

text: signiﬁcantly

text: degrades

text: the

text: detection

text: performance,

text: one

text: community.

text: compared

text: to

text: our

text: original

text: KroMFac

text: framework.

text: The

text: number

text: of

text: communities

text: per

text: node

text: indicates

text: the

text: EXPERIMENTAL

text: EVALUATION

text: average

text: number

text: of

text: communities

text: to

text: which

text: each

text: of

text: the

text: In

text: this

text: section,

text: we

text: ﬁrst

text: describe

text: both

text: synthetic

text: and

text: overlapping

text: nodes

text: belongs.

text: real-world

text: datasets.

text: By

text: adopting

text: the

text: NMI

text: as

text: a

text: popular

text: To

text: cover

text: various

text: domains

text: of

text: network

text: applications,

text: we

text: gen-

text: information-theoretic

text: performance

text: metric,

text: we

text: then

text: present

text: erate

text: ﬁve

text: LFR

text: graphs

text: with

text: differing

text: parameter

text: settings

text: as

text: the

text: performance

text: of

text: our

text: community

text: detection

text: framework

text: speciﬁed

text: in

text: Table

text: and

text: compare

text: it

text: against

text: the

text: two

text: baseline

text: schemes.

text: Datasets

text: Real-World

text: Datasets

text: To

text: evaluate

text: the

text: community

text: detection

text: performance

text: of

text: To

text: validate

text: the

text: applicability

text: of

text: our

text: approach,

text: two

text: types

text: our

text: approach,

text: we

text: rely

text: on

text: datasets

text: for

text: which

text: ground-truth

text: of

text: real-world

text: datasets

text: are

text: also

text: used

text: for

text: evaluation.

text: More

text: communities

text: are

text: explicitly

text: labeled.

text: In

text: the

text: following,

text: both

text: speciﬁcally,

text: from

text: the

text: available

text: SNAP

text: datasets

text: we

text: use

text: synthetic

text: and

text: real-world

text: datasets

text: across

text: various

text: domains

text: the

text: Amazon

text: product

text: co-purchasing

text: network

text: and

text: the

text: are

text: taken

text: into

text: account.

text: collaboration

text: network

text: of

text: DBLP

text: The

text: statistics

text: of

text: these

text: Synthetic

text: Datasets

text: datasets

text: are

text: summarized

text: in

text: Table

text: and

text: the

text: basic

text: character-

text: We

text: construct

text: synthetic

text: graphs

text: via

text: the

text: extended

text: istics

text: of

text: each

text: network

text: are

text: described

text: in

text: the

text: following:

text: Lancichinetti-Fortunato-Radicchi

text: (LFR)

text: benchmark

text: The

text: number

text: of

text: nodes

text: (N.N.):

text: In

text: the

text: Amazon

text: network,

text: which

text: is

text: built

text: upon

text: a

text: generative

text: model

text: that

text: creates

text: nodes

text: nodes

text: represent

text: products.

text: In

text: DBLP,

text: nodes

text: represent

text: along

text: with

text: prior

text: known

text: community

text: labels.

text: The

text: benchmark

text: authors.

text: is

text: capable

text: of

text: generating

text: graphs

text: that

text: replicate

text: important

text: The

text: number

text: of

text: edges

text: (N.E.):

text: In

text: the

text: Amazon

text: network,

text: features

text: of

text: real

text: social

text: networks

text: such

text: as

text: the

text: power-law

text: edges

text: connect

text: products

text: that

text: are

text: commonly

text: pur-

text: ##LTLine##

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text: (cid:104) (cid:105) a˙ (u) = − (cid:96)2p ψ˜˙ (u, r)2 − 2ψ¨˜ (u, r) , r ≥ r (u) (58) 0 0 0 s 24π (cid:90) in which a is the Bondi mass of the whole black hole, and 0 (cid:48) − (cid:48) (cid:48) rs(u) ee a r (cid:48) ψ˜0(u) = (1 + V )r(cid:48)2 − 2ar(cid:48) + e2 dr (59) 0 Due to the conservation of charges on each shell, de ⇒ α = 0 De(u, r) = 0 (60) du α − At this point, if one imposes the null matter condition of V (u, r) = 1F (u, r), the 2 geometry and the dynamics will be completely determined in a consistent way by (47) (49), (56), (57) and (60). 4 The Asymptotic Analysis In this section, we analyze the behavior of continuously collapsing RNV black hole un- der various circumstances. As an assumption, every shell obeys the null condition r˙ = − α 1F (u , r ). We will consider the case in which every collapsing shell is near its apparent 2 α α α | | (cid:28) horizon. The shell being near its apparent horizon implies that r˙ 1 and the dependence α of time in a (u ) cannot be ignored. α α The discussion will be divided into two parts. The ﬁrst one is the non-extremal case in which a is not too close to e , in which (23) can hold. We also consider a region of 0 0 near extremal inner layers covered by a region of collapsing neutral outer layers. Under this situation, quasi-static approximation can be applied. In the second part, we consider the near-extremal case, in which every shell is near its extremal limit in the initial condition. In this case, (23) does not hold. A more sophisticated treatment is needed under this situation. 4.1 The Non-extremal Behavior Being parallel to the notation used in [5], we deﬁne d dU ˜˙ ξ(u) := log( ) = ψ . (61) 0 du du (cid:18) (cid:19) (cid:90) (cid:18)− (cid:19) By inserting (59), the deﬁnition of ψ˜ , into (61) with V = 1F , we get 0 2 − (cid:48) (cid:48) (cid:48) − (cid:48) r˙ 2a (u, r ) 2e (u, r )e rs(u) d ee a r ξ = s s + s 0 + 2 − dr. (62) F r r2 du r2 2ar + e2 s s 0 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: a˙

text: (u)

text: ψ˜˙

text: (u,

text: 2ψ¨˜

text: (u,

text: r)

text: r

text: r

text: (u)

text: s

text: in

text: which

text: a

text: is

text: the

text: Bondi

text: mass

text: of

text: the

text: whole

text: black

text: hole,

text: and

text: rs(u)

text: ee

text: a

text: r

text: V

text: dr

text: Due

text: to

text: the

text: conservation

text: of

text: charges

text: on

text: each

text: shell,

text: de

text: α

text: De(u,

text: r)

text: du

text: α

text: At

text: this

text: point,

text: if

text: one

text: imposes

text: the

text: matter

text: condition

text: of

text: V

text: (u,

text: r)

text: (u,

text: r),

text: the

text: geometry

text: and

text: the

text: dynamics

text: will

text: be

text: completely

text: determined

text: in

text: a

text: consistent

text: way

text: by

text: and

text: The

text: Asymptotic

text: Analysis

text: In

text: this

text: section,

text: we

text: analyze

text: the

text: behavior

text: of

text: continuously

text: collapsing

text: RNV

text: black

text: hole

text: un-

text: der

text: various

text: circumstances.

text: As

text: an

text: assumption,

text: every

text: shell

text: obeys

text: the

text: condition

text: r˙

text: α

text: (u

text: r

text: We

text: will

text: consider

text: the

text: case

text: in

text: which

text: every

text: collapsing

text: shell

text: is

text: near

text: its

text: apparent

text: α

text: α

text: α

text: horizon.

text: The

text: shell

text: being

text: near

text: its

text: apparent

text: horizon

text: implies

text: that

text: r˙

text: and

text: the

text: dependence

text: α

text: of

text: time

text: in

text: a

text: (u

text: cannot

text: be

text: ignored.

text: α

text: α

text: The

text: discussion

text: will

text: be

text: divided

text: into

text: two

text: parts.

text: The

text: ﬁrst

text: one

text: is

text: the

text: non-extremal

text: case

text: in

text: which

text: a

text: is

text: not

text: too

text: close

text: to

text: e

text: in

text: which

text: can

text: hold.

text: We

text: also

text: consider

text: a

text: region

text: of

text: near

text: extremal

text: inner

text: layers

text: covered

text: by

text: a

text: region

text: of

text: collapsing

text: neutral

text: outer

text: layers.

text: Under

text: this

text: situation,

text: quasi-static

text: approximation

text: can

text: be

text: applied.

text: In

text: the

text: second

text: part,

text: we

text: consider

text: the

text: near-extremal

text: case,

text: in

text: which

text: every

text: shell

text: is

text: near

text: its

text: extremal

text: limit

text: in

text: the

text: initial

text: condition.

text: In

text: this

text: case,

text: does

text: not

text: hold.

text: A

text: more

text: sophisticated

text: treatment

text: is

text: needed

text: under

text: this

text: situation.

text: The

text: Non-extremal

text: Behavior

text: Being

text: parallel

text: to

text: the

text: notation

text: used

text: in

text: we

text: deﬁne

text: d

text: dU

text: ξ(u)

text: log(

text: ψ

text: du

text: du

text: By

text: inserting

text: the

text: deﬁnition

text: of

text: ψ˜

text: into

text: with

text: V

text: we

text: get

text: r˙

text: (u,

text: r

text: (u,

text: r

text: )e

text: rs(u)

text: d

text: ee

text: a

text: r

text: ξ

text: s

text: s

text: s

text: dr.

text: F

text: r

text: du

text: s

text: s

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: − Figure 3: The plot of Hawking temperature against a e . All the units are Planck units. 0 0 4.4 The Energy Momentum Tensor We shall calculate the energy momentum tensor T by inserting the RNV metric (49) µν into Einstein tensor. The results are: eψ − − G = (r∂ F eψF (F 1 + r∂ F )) (100) uu r2 u r −eψ − G = (F 1 + r∂ F ) (101) ur r2 r (cid:18) (cid:19) 2∂ ψ r G = (102) rr r 3 1 − − G = r(∂ F + F ∂ ψ) + r2 ∂ F ∂ ψ + F (∂ ψ)2 + ∂2F + F ∂2ψ e ψ∂ ∂ ψ (103) θθ r r 2 r r r 2 r r r u G = sin2 θG (104) φφ θθ All the other components of Einstein tensor are zero. Incorporating the condition of null infalling matter and obtaining from (48) (cid:48) − (cid:48) (cid:48) ra ee ψ = 2 , (105) r2F 22 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Figure

text: The

text: plot

text: of

text: Hawking

text: temperature

text: against

text: a

text: e

text: All

text: the

text: units

text: are

text: Planck

text: units.

label: section

text: The

text: Energy

text: Momentum

text: Tensor

text: We

text: shall

text: calculate

text: the

text: energy

text: momentum

text: tensor

text: T

text: by

text: inserting

text: the

text: RNV

text: metric

text: µν

text: into

text: Einstein

text: tensor.

text: The

text: results

text: are:

text: eψ

text: G

text: (r∂

text: F

text: eψF

text: (F

text: r∂

text: F

text: uu

text: u

text: r

text: −eψ

text: G

text: (F

text: r∂

text: F

text: ur

text: r

text: 2∂

text: ψ

text: r

text: G

text: rr

text: r

text: G

text: r(∂

text: F

text: F

text: ψ)

text: F

text: ψ

text: F

text: F

text: ∂2ψ

text: e

text: ψ∂

text: ψ

text: θθ

text: r

text: r

text: r

text: r

text: r

text: r

text: r

text: r

text: u

text: G

text: θG

text: φφ

text: θθ

text: All

text: the

text: other

text: components

text: of

text: Einstein

text: tensor

text: are

text: zero.

text: Incorporating

text: the

text: condition

text: of

text: infalling

text: matter

text: and

text: obtaining

text: from

text: ra

text: ee

text: ψ

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 Then, the extremal condition δΩ/(cid:90)δm(x) = 0 gives the fol(cid:90)lowing in(cid:90)tegral equation: − m + m(x) m d3p d3p (cid:48) (cid:48) (cid:48) (cid:48) 0 + trG(p; x, x) + dx trG(p; x, x )m(x )G(p; x , x) = 0, (25) 2G (2π)3 (2π)3 where trG(p; x, x) ca(cid:16)n be evaluated(cid:17)by follow(cid:16) ing the property (cid:17)(cid:104) (cid:105) − −1 (cid:104) − − − (cid:105) −1 tr iD/ + µγ0 m = tr i(ωl + iµ)γ0 iD/i m (ωl + iµ)2 D2i + m2 + qfσµνFµν − − −1 = m tr (ωl + iµ)2 D2i + m2 + qfσµνFµν (26) as [29, 30]: (cid:88) −2m 2 Γ(1 + 2a)Γ(1 + 2b) trG(p; x, x) = W ± g1(s, x)g2(s, x), W = λ Γ(a + b + 1 − c )Γ(a + b + 1 + c ), s= 2 s 2 s (cid:16) (cid:17) − 1 − 1 g1(s, x) = ξa(1 ξ)bF(a + b + 2 cs, a + b + 2 + cs; 1 + 2a; ξ), (cid:113) − 1 − 1 (cid:113) − 1 + tanh x g2(s, x) = ξa(1 ξ)bF(a + b + 2 cs, a + b + 2 + cs; 1 + 2b; 1 ξ), ξ = 2 λ λ − λ a = 2 (p2 qfBλ)2 + (ωl + iµ)2 + p23 + m2, b = 2 (p2 + qfBλ)2 + (ωl + iµ)2(cid:113)+ p23 + m2. (27) → − → − − − In the limit B 0, F(a+b+12 cs, a+b+12+cs; 1+2a; ξ) (1 ξ) 2a and the trace reduces to 2m/ p22 + (ωl + iµ)2 + p23 +→m∞2, which is consistent with the usual one obtained in energy-momentum space but p is integrated over ﬁrst here. In the limit x → 1 or ξ 1, the hypergeometric functions become − − 1 − 1 (1 ξ) 2b Γ(1 + 2a)Γ(1 + 2b) F(a + b + 2 cs, a + b + 2 + cs; 1 + 2a; ξ) = 2b Γ(a + b + 1 − c )Γ(a + b + 1 + c ), 2 s 2 s 1 − 1 − F(a + b + 2 cs, a + b + 2 + cs; 1 + 2b; 1 ξ) = 1. (28) Then, trG(p; x, x) becomes magnetic ﬁeld independent after shifting the integral variable p2 in b which indicates lim →∞ m(x) = 0 as expected. For the last term on the left-hand side of the gap equation Eq.(25), the eﬀective integral re- x gion is constrained by m(x) or originally by B(x) to order λ. Thus, for not too large λ, it is enough to evaluate this term with the fermion propagator in the absence of magnetic ﬁeld because that only gives next-to-next-to-next order contribution, and we can (cid:48) ≈ just take m(x ) m(x) as its leading order contribution of the Taylor expansions around x. Finally, the integral equation Eq.(25) can be reduced to an algeb(cid:34)ra equa(cid:90)tion: (cid:35) (cid:34)(cid:90) (cid:35) (cid:16) (cid:17) − − 1 d4p − −2 1 d3p − → m(x) = 2G + (2π)4 tr i(ωl + iµ)γ0 + piγi m (2π)3 trG(p; x, x) (B 0) . (29) The prefactor in the expression of m(x) is actually the propagator of σ mode at vanishing energy-momentum and can be given d(cid:34)irectly(cid:90)as [34]: (cid:35)  (cid:90) (cid:90)  (cid:16) (cid:17)  (cid:88)  1 d4p − −2 −1 1 − Λ dp p4 ∞ dp p4 1 −1 2G + (2π)4 tr i(ωl + iµ)γ0 + piγi m = 2G NcNf 0 π2 E3(p) +NcNf s=± 0 π2 E3(p) 1 + e(E(p)+sµ)/T .(30) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: Then,

text: the

text: extremal

text: condition

font: UPTIMB+rtxmi

font: JPOSER+rtxr

text: gives

text: the

text: equation:

font: UECPNB+txsy

text: m

text: m(x)

text: m

text: trG(p;

text: x,

text: x)

text: dx

text: trG(p;

text: x,

text: x

text: )m(x

text: )G(p;

text: x

text: x)

text: where

text: trG(p;

text: x,

text: x)

text: be

text: ing

text: the

text: property

font: GLIDDI+txex

text: −1

text: −1

text: tr

text: iD/

text: µγ0

text: m

text: tr

text: i(ωl

text: iD/i

text: m

text: (ωl

text: qfσµνFµν

text: −1

text: m

text: tr

text: (ωl

text: qfσµνFµν

text: as

text: trG(p;

text: x,

text: x)

text: W

text: x),

text: W

text: λ

text: Γ(a

text: b

text: c

text: )Γ(a

text: b

text: c

text: s=

text: s

text: s

text: x)

text: ξ)bF(a

text: b

text: cs,

text: a

text: b

text: cs;

text: ξ),

text: tanh

text: x

text: x)

text: ξ)bF(a

text: b

text: cs,

text: a

text: b

text: cs;

text: ξ),

text: ξ

text: λ

text: λ

text: λ

text: a

text: (ωl

text: b

text: (ωl

text: In

text: the

text: limit

text: B

text: cs,

text: ξ)

text: ξ)

text: and

text: the

text: trace

text: reduces

text: to

text: (ωl

text: which

text: is

text: consistent

text: with

text: the

text: usual

text: one

text: obtained

text: in

text: energy-momentum

text: space

text: but

text: p

text: is

text: integrated

text: over

text: ﬁrst

text: here.

text: In

text: the

text: limit

text: x

text: or

text: ξ

text: the

text: hypergeometric

text: functions

text: become

text: ξ)

text: F(a

text: b

text: cs,

text: a

text: b

text: cs;

text: ξ)

text: Γ(a

text: b

text: c

text: )Γ(a

text: b

text: c

text: s

text: s

text: F(a

text: b

text: cs,

text: a

text: b

text: cs;

text: ξ)

text: Then,

text: trG(p;

text: x,

text: x)

text: becomes

text: magnetic

text: ﬁeld

text: independent

text: after

text: shifting

text: the

text: integral

text: variable

text: in

text: b

text: which

text: indicates

text: lim

text: m(x)

text: as

text: expected.

text: For

text: the

text: last

text: term

text: on

text: the

text: left-hand

text: side

text: of

text: the

text: gap

text: equation

text: the

text: eﬀective

text: integral

text: re-

text: x

text: gion

text: is

text: constrained

text: by

text: m(x)

text: or

text: originally

text: by

text: B(x)

text: to

text: order

text: λ.

text: Thus,

text: for

text: not

text: too

text: large

text: λ,

text: it

text: is

text: enough

text: to

text: evaluate

text: this

text: term

text: with

text: the

text: fermion

text: propagator

text: in

text: the

text: absence

text: of

text: magnetic

text: ﬁeld

text: because

text: that

text: only

text: gives

text: next-to-next-to-next

text: order

text: contribution,

text: and

text: we

text: can

text: just

text: take

text: m(x

text: m(x)

text: as

text: its

text: leading

text: order

text: contribution

text: of

text: the

text: Taylor

text: expansions

text: around

text: x.

text: Finally,

text: the

text: integral

text: equation

text: can

text: be

text: reduced

text: to

text: an

text: −2

text: m(x)

text: tr

text: i(ωl

text: piγi

text: m

text: trG(p;

text: x,

text: x)

text: (B

text: The

text: prefactor

text: in

text: the

text: expression

text: of

text: m(x)

text: is

text: actually

text: the

text: propagator

text: of

text: σ

text: mode

text: at

text: vanishing

text: energy-momentum

text: and

text: can

text: be

text: given

text: −2

text: −1

text: Λ

text: dp

text: dp

text: −1

text: tr

text: i(ωl

text: piγi

text: m

text: NcNf

text: π2

text: +NcNf

text: s=±

text: π2

text: e(E(p)+sµ)/T

text: ##LTLine##

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text: 9 is illuminated, the ﬂuctuation is not sensitive to the change IV. CONCLUSIONS of m when it is still considerably large and is deeply sup- pressed when it becomes small, which justiﬁes the Taylor expansions in the whole regions. Because of the approx- imate chiral symmetry with ﬁnite current quark mass and non-renormalizability with four fermion couplings of the NJL In this work, I ﬁrst developed a formalism to evaluate the model, the expected features across the phase transition for thermodynamic potential with constant fermion mass in the 2 + 1 dimensions are not found. Specially, the vanishing of region where magnetic ﬁeld is localized and also for the sys- m(x) is not found across the ﬁrst-order transition because m is tem with ﬁxed size under the framework of NJL model. Then still ﬁnite after the transition. apart from the magnetic ﬁeld independent terms, the contribu- tions from the discrete and continuum spectra are compared 0.30 with each other, which indicates the negligible of the latter. Finally, we tried to study the local chiral symmetry breaking 0.25 due to weak semi-localized magnetic ﬁeld, which is the main 0.20 motivation of this work, by using Taylor expansion technique. 0.15 0.10 The main ﬁndings are the followings. In the constant m 0.05 ansatz, both the magnetic ﬁeld magnitude B and scale λ tend to catalyze chiral symmetry breaking in 2 + 1 and 3 + 1 di- 0.00 0.10 0.12 0.14 0.16 0.18 0.20 mensional cases. And in the weak magnetic ﬁeld approxi- mation, the local chiral symmetry breaking is also found to T(cid:144)GeV be enhanced by both B and λ which conﬁrms the qualitative 0.30 features from the constant m ansatz. Thus, the results indi- cate the importance of inhomogeneous magnetic ﬁeld eﬀect 0.25 in HICs with much larger B and that the expanding of ﬁreball (λ becomes larger) doesn’t necessarily reduce the magnetic 0.20 ﬁeld eﬀect during the period when B sustains. Furthermore, 0.15 the mass ﬂuctuation m(x) is found to be not sensitive to the change of temperature T or baryon chemical potential µ when 0.10 the global mass is still considerably large, which further sup- ports the importance of the inhomogeneous magnetic eﬀect in 0.05 HICs. 0.00 0.30 0.32 0.34 0.36 0.38 0.40 Μ(cid:144)GeV Acknowledgments— I thank Xu-guang Huang from Fudan FIG. 7: (color online) The constant mass gap m (black solid lines) and the maximal ﬂuctuation m(0) (red dashed lines) as functions of University for his comments on this work. GC is supported temperature T at vanishing baryon chemical potential (upper panel) by the Thousand Young Talents Program of China, Shanghai and baryon chemical potential µ at vanishing temperature in 3 + 1 Natural Science Foundation with Grant No. 14ZR1403000, dimensional NJL model. The magnitude and scale of the magnetic NSFC with Grant No. 11535012 and No. 11675041, | | − ﬁeld are chosen as ( eB 1/2,λ) = (0.3GeV,2GeV 1). and China Postdoctoral Science Foundation with Grant No. KLH1512072. [1] V. Skokov, A. Y. Illarionov and V. Toneev, “Estimate of the [4] W. T. Deng and X. G. Huang, “Event-by-event generation of magnetic ﬁeld strength in heavy-ion collisions,” Int. J. Mod. electromagnetic ﬁelds in heavy-ion collisions,” Phys. Rev. C Phys. A 24, 5925 (2009). 85, 044907 (2012). [2] V. Voronyuk, V. D. Toneev, W. Cassing, E. L. Bratkovskaya, [5] G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, V. P. Konchakovski and S. A. Voloshin, “(Electro-)Magnetic S. Krieg, A. Schafer and K. K. Szabo, “The QCD phase diagram ﬁeld evolution in relativistic heavy-ion collisions,” Phys. Rev. for external magnetic ﬁelds,” JHEP 1202, 044 (2012). C 83, 054911 (2011). [6] G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz [3] A. Bzdak and V. Skokov, “Event-by-event ﬂuctuations of mag- and A. Schafer, “QCD quark condensate in external magnetic netic and electric ﬁelds in heavy ion collisions,” Phys. Lett. B ﬁelds,” Phys. Rev. D 86, 071502 (2012). 710, 171 (2012). [7] F. Bruckmann, G. Endrodi and T. G. Kovacs, “Inverse magnetic VeG(cid:144)L0Hm&m VeG(cid:144)L0Hm&m ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: is

text: illuminated,

text: the

text: ﬂuctuation

text: is

text: not

text: sensitive

text: to

text: the

text: change

text: IV.

text: CONCLUSIONS

text: of

text: m

text: when

text: it

text: is

text: still

text: considerably

text: large

text: and

text: is

text: deeply

text: sup-

text: pressed

text: when

text: it

text: becomes

text: small,

text: which

text: justiﬁes

text: the

text: Taylor

text: expansions

text: in

text: the

text: whole

text: regions.

text: Because

text: of

text: the

text: approx-

text: imate

text: chiral

text: symmetry

text: with

text: ﬁnite

text: current

text: quark

text: mass

text: and

text: non-renormalizability

text: with

text: four

text: fermion

text: couplings

text: of

text: the

text: NJL

text: In

text: this

text: work,

text: I

text: ﬁrst

text: developed

text: a

text: formalism

text: to

text: evaluate

text: the

text: model,

text: the

text: expected

text: features

text: across

text: the

text: phase

text: transition

text: for

text: thermodynamic

text: potential

text: with

text: constant

text: fermion

text: mass

text: in

text: the

text: dimensions

text: are

text: not

text: found.

text: Specially,

text: the

text: vanishing

text: of

text: region

text: where

text: magnetic

text: ﬁeld

text: is

text: localized

text: and

text: also

text: for

text: the

text: sys-

text: m(x)

text: is

text: not

text: found

text: across

text: the

text: ﬁrst-order

text: transition

text: because

text: m

text: is

text: tem

text: with

text: ﬁxed

text: size

text: under

text: the

text: framework

text: of

text: NJL

text: model.

text: Then

text: still

text: ﬁnite

text: after

text: the

text: transition.

text: apart

text: from

text: the

text: magnetic

text: ﬁeld

text: independent

text: terms,

text: the

text: contribu-

text: tions

text: from

text: the

text: discrete

text: and

text: continuum

text: spectra

text: are

text: compared

text: with

text: each

text: other,

text: which

text: indicates

text: the

text: negligible

text: of

text: the

text: latter.

text: Finally,

text: we

text: tried

text: to

text: study

text: the

text: local

text: chiral

text: symmetry

text: breaking

text: due

text: to

text: weak

text: semi-localized

text: magnetic

text: ﬁeld,

text: which

text: is

text: the

text: main

text: motivation

text: of

text: this

text: work,

text: by

text: using

text: Taylor

text: expansion

text: technique.

text: The

text: main

text: ﬁndings

text: are

text: the

text: followings.

text: In

text: the

text: constant

text: m

text: ansatz,

text: both

text: the

text: magnetic

text: ﬁeld

text: magnitude

text: B

text: and

text: scale

text: λ

text: tend

text: to

text: catalyze

text: chiral

text: symmetry

text: breaking

text: in

text: and

text: di-

text: mensional

text: cases.

text: And

text: in

text: the

text: weak

text: magnetic

text: ﬁeld

text: approxi-

text: mation,

text: the

text: local

text: chiral

text: symmetry

text: breaking

text: is

text: also

text: found

text: to

text: be

text: enhanced

text: by

text: both

text: B

text: and

text: λ

text: which

text: conﬁrms

text: the

text: qualitative

text: features

text: from

text: the

text: constant

text: m

text: ansatz.

text: Thus,

text: the

text: results

text: indi-

text: cate

text: the

text: importance

text: of

text: inhomogeneous

text: magnetic

text: ﬁeld

text: eﬀect

text: in

text: HICs

text: with

text: much

text: larger

text: B

text: and

text: that

text: the

text: expanding

text: of

text: ﬁreball

text: (λ

text: becomes

text: larger)

text: doesn’t

text: necessarily

text: reduce

text: the

text: magnetic

text: ﬁeld

text: eﬀect

text: during

text: the

text: period

text: when

text: B

text: sustains.

text: Furthermore,

text: the

text: mass

text: ﬂuctuation

text: m(x)

text: is

text: found

text: to

text: be

text: not

text: sensitive

text: to

text: the

text: change

text: of

text: temperature

text: T

text: or

text: baryon

text: chemical

text: potential

text: µ

text: when

text: the

text: global

text: mass

text: is

text: still

text: considerably

text: large,

text: which

text: further

text: sup-

text: ports

text: the

text: importance

text: of

text: the

text: inhomogeneous

text: magnetic

text: eﬀect

text: in

text: HICs.

text: Acknowledgments—

text: I

text: thank

text: Xu-guang

text: Huang

text: from

text: Fudan

text: FIG.

text: (color

text: online)

text: The

text: constant

text: mass

text: gap

text: m

text: (black

text: solid

text: lines)

text: and

text: the

text: maximal

text: ﬂuctuation

text: (red

text: dashed

text: lines)

text: as

text: functions

text: of

text: University

text: for

text: his

text: comments

text: on

text: this

text: work.

text: GC

text: is

text: supported

text: temperature

text: T

text: at

text: vanishing

text: baryon

text: chemical

text: potential

text: (upper

text: panel)

text: by

text: the

text: Thousand

text: Young

text: Talents

text: Program

text: of

text: China,

text: Shanghai

text: and

text: baryon

text: chemical

text: potential

text: µ

text: at

text: vanishing

text: temperature

text: in

text: Natural

text: Science

text: Foundation

text: with

text: Grant

text: No.

text: dimensional

text: NJL

text: model.

text: The

text: magnitude

text: and

text: scale

text: of

text: the

text: magnetic

text: NSFC

text: with

text: Grant

text: No.

text: and

text: No.

text: ﬁeld

text: are

text: chosen

text: as

text: eB

text: and

text: China

text: Postdoctoral

text: Science

text: Foundation

text: with

text: Grant

text: No.

text: V.

text: Skokov,

text: A.

text: Y.

text: Illarionov

text: and

text: V.

text: Toneev,

text: “Estimate

text: of

text: the

text: W.

text: T.

text: Deng

text: and

text: X.

text: G.

text: Huang,

text: “Event-by-event

text: generation

text: of

text: magnetic

text: ﬁeld

text: strength

text: in

text: heavy-ion

text: collisions,”

text: Int.

text: J.

text: Mod.

text: electromagnetic

text: ﬁelds

text: in

text: heavy-ion

text: collisions,”

text: Phys.

text: Rev.

text: C

text: Phys.

text: A

text: V.

text: Voronyuk,

text: V.

text: D.

text: Toneev,

text: W.

text: Cassing,

text: E.

text: L.

text: Bratkovskaya,

text: G.

text: S.

text: Bali,

text: F.

text: Bruckmann,

text: G.

text: Endrodi,

text: Z.

text: Fodor,

text: S.

text: D.

text: Katz,

text: V.

text: P.

text: Konchakovski

text: and

text: S.

text: A.

text: Voloshin,

text: “(Electro-)Magnetic

text: S.

text: Krieg,

text: A.

text: Schafer

text: and

text: K.

text: K.

text: Szabo,

text: “The

text: QCD

text: phase

text: diagram

text: ﬁeld

text: evolution

text: in

text: relativistic

text: heavy-ion

text: collisions,”

text: Phys.

text: Rev.

text: for

text: external

text: magnetic

text: ﬁelds,”

text: JHEP

text: C

text: G.

text: S.

text: Bali,

text: F.

text: Bruckmann,

text: G.

text: Endrodi,

text: Z.

text: Fodor,

text: S.

text: D.

text: Katz

text: A.

text: Bzdak

text: and

text: V.

text: Skokov,

text: “Event-by-event

text: ﬂuctuations

text: of

text: mag-

text: and

text: A.

text: Schafer,

text: “QCD

text: quark

text: condensate

text: in

text: external

text: magnetic

text: netic

text: and

text: electric

text: ﬁelds

text: in

text: heavy

text: ion

text: collisions,”

text: Phys.

text: Lett.

text: B

text: ﬁelds,”

text: Phys.

text: Rev.

text: D

text: F.

text: Bruckmann,

text: G.

text: Endrodi

text: and

text: T.

text: G.

text: Kovacs,

text: “Inverse

text: magnetic

text: ##LTFigure##

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text: 3 FIG. 2. (Color online) (001) surface states of Co Sn S for S- and Sn-terminals. (a) Sn-terminated surface FS with energy ﬁxed 3 2 2 at the Weyl points. (b) Energy dispersion for the Sn-terminated surface along a k-path crossing a pair of Weyl points connected by a Fermi arc. (c) Sn-terminated surface FS at the charge neutral point. (d) Schematic diagram of the FS distribution on the Sn-terminated surface. (e) S-terminated surface FS with energy ﬁxed at the Weyl points. (f) Surface band structure for S-terminated states taken along the same direction as that in (b). (g) S-terminated surface FS at the Fermi energy, E . (h) F Schematic diagram of the FS distribution on the S-terminated surface. The dashed curves in (d) and (h) represent surface states that merge into the bulk. The Fermi arcs and trivial FSs are shown in yellow and black, respectively. properties. measurement. A typical feature for WSMs is the non-closed Fermi In order to check for possible Fermi arc states at E , F arc surface state. For this quasi-2D lattice structure in we also analyzed the energy dispersion of the Fermi arc the xy-plane, it is easily to obtain the (001) surface in related states in energy space. As shown in Fig. 2(b), experiments. So there are two possible cleaving planes, the Fermi arc related bands can extend to E -0.1 eV F breaking the S–Co bonds or the Sn–S bonds. On account from the energy of Weyl points (E ), which oﬀers good W of the atomic separation in the z-direction, we found that opportunity for detecting the Fermi arcs in the low energy the bonding between the Sn and S layers is much weaker, region. The corresponding FS is shown in Fig. 2(c), and this was conﬁrmed by our total energy calculations. where the Fermi arcs are located around the corners The formation energy required to break the Sn–S bonds of the BZ, with the K¯ -points situated at the center is around 2.9 eV per cell. Therefore the Sn–S bonds are of the three Fermi arcs. For the S-terminated state, easier to cleave. Thus we get two diﬀerent terminations most of the Fermi arcs merge into the bulk, making for the cleaved (001) surface, i.e, S- and Sn-terminals, the visible Fermi arc states much shorter than those respectively. exhibited for the Sn-terminated states, see Fig. 2(e, h). Moreover, the Fermi arc related bands extend to the Fig. 2 shows the surface FSs and energy dispersion higher energy region above the Weyl points, as indicated corresponding to the two diﬀerently terminated surfaces. in Fig. 2(f). Therefore, to observe the Fermi arcs from Fixing the energy at the Weyl points for the Sn-terminated the S-terminal by ARPES, an electron-doped sample and surface, one can easily ﬁnd the single surface FS starting surface electron-doping is necessary. However, we note from one Weyl point to another Weyl point with the that although potassium doping can be used to achieve opposite chirality, which is the typical expected behavior small upward shifts in the surface chemical potential50, of Fermi arcs. Taking the C rotation symmetry into 3z this method does not work well in every situation. account, there are three such Fermi arcs in the ﬁrst BZ, as indicated in Fig. 2(a, d). Although a little part of Moreover, in a signiﬁcant diﬀerence to WSMs that the Fermi arc merging into the bulk, the Fermi arc still exhibit time-reversal symmetry, the surface state measurements extends to about 25% of the reciprocal lattice vector, of Co Sn S need to consider the eﬀects of magnetic 3 2 2 which is suﬃciently long to be detected by an ARPES domains. The size of these domains are typically around ##LTFigure##

text: FIG.

text: (Color

text: online)

text: surface

text: states

text: of

text: Co

text: Sn

text: S

text: for

text: S-

text: and

text: Sn-terminals.

text: (a)

text: Sn-terminated

text: surface

text: FS

text: with

text: energy

text: ﬁxed

text: at

text: the

text: Weyl

text: points.

text: (b)

text: Energy

text: dispersion

text: for

text: the

text: Sn-terminated

text: surface

text: along

text: a

text: k-path

text: crossing

text: a

text: pair

text: of

text: Weyl

text: points

text: connected

text: by

text: a

text: Fermi

text: arc.

text: (c)

text: Sn-terminated

text: surface

text: FS

text: at

text: the

text: charge

text: neutral

text: point.

text: (d)

text: Schematic

text: diagram

text: of

text: the

text: FS

text: distribution

text: on

text: the

text: Sn-terminated

text: surface.

text: (e)

text: S-terminated

text: surface

text: FS

text: with

text: energy

text: ﬁxed

text: at

text: the

text: Weyl

text: points.

text: (f)

text: Surface

text: band

text: structure

text: for

text: S-terminated

text: states

text: taken

text: along

text: the

text: same

text: direction

text: as

text: that

text: in

text: (b).

text: (g)

text: S-terminated

text: surface

text: FS

text: at

text: the

text: Fermi

text: energy,

text: E

text: (h)

text: F

text: Schematic

text: diagram

text: of

text: the

text: FS

text: distribution

text: on

text: the

text: S-terminated

text: surface.

text: The

text: dashed

text: curves

text: in

text: (d)

text: and

text: (h)

text: represent

text: surface

text: states

text: that

text: merge

text: into

text: the

text: bulk.

text: The

text: Fermi

text: arcs

text: and

text: trivial

text: FSs

text: are

text: shown

text: in

text: yellow

text: and

text: black,

text: respectively.

text: properties.

text: measurement.

text: A

text: typical

text: feature

text: for

text: WSMs

text: is

text: the

text: non-closed

text: Fermi

text: In

text: order

text: to

text: check

text: for

text: possible

text: Fermi

text: arc

text: states

text: at

text: E

text: F

text: arc

text: surface

text: state.

text: For

text: this

text: lattice

text: structure

text: in

text: we

text: also

text: analyzed

text: the

text: energy

text: dispersion

text: of

text: the

text: Fermi

text: arc

text: the

text: xy-plane,

text: it

text: is

text: easily

text: to

text: obtain

text: the

text: surface

text: in

text: related

text: states

text: in

text: energy

text: space.

text: As

text: shown

text: in

text: Fig.

text: experiments.

text: So

text: there

text: are

text: two

text: possible

text: cleaving

text: planes,

text: the

text: Fermi

text: arc

text: related

text: bands

text: can

text: extend

text: to

text: E

text: eV

text: F

text: breaking

text: the

text: S–Co

text: bonds

text: or

text: the

text: Sn–S

text: bonds.

text: On

text: account

text: from

text: the

text: energy

text: of

text: Weyl

text: points

text: (E

text: which

text: oﬀers

text: good

text: W

text: of

text: the

text: atomic

text: separation

text: in

text: the

text: z-direction,

text: we

text: found

text: that

text: opportunity

text: for

text: detecting

text: the

text: Fermi

text: arcs

text: in

text: the

text: low

text: energy

text: the

text: bonding

text: between

text: the

text: Sn

text: and

text: S

text: layers

text: is

text: much

text: weaker,

text: region.

text: The

text: corresponding

text: FS

text: is

text: shown

text: in

text: Fig.

text: and

text: this

text: was

text: conﬁrmed

text: by

text: our

text: total

text: energy

text: calculations.

text: where

text: the

text: Fermi

text: arcs

text: are

text: located

text: around

text: the

text: corners

text: The

text: formation

text: energy

text: required

text: to

text: break

text: the

text: Sn–S

text: bonds

text: of

text: the

text: BZ,

text: with

text: the

text: K¯

text: -points

text: situated

text: at

text: the

text: center

text: is

text: around

text: eV

text: per

text: cell.

text: Therefore

text: the

text: Sn–S

text: bonds

text: are

text: of

text: the

text: three

text: Fermi

text: arcs.

text: For

text: the

text: S-terminated

text: state,

text: easier

text: to

text: cleave.

text: Thus

text: we

text: get

text: two

text: diﬀerent

text: terminations

text: most

text: of

text: the

text: Fermi

text: arcs

text: merge

text: into

text: the

text: bulk,

text: making

text: for

text: the

text: cleaved

text: surface,

text: i.e,

text: S-

text: and

text: Sn-terminals,

text: the

text: visible

text: Fermi

text: arc

text: states

text: much

text: shorter

text: than

text: those

text: respectively.

text: exhibited

text: for

text: the

text: Sn-terminated

text: states,

text: see

text: Fig.

text: h).

text: Moreover,

text: the

text: Fermi

text: arc

text: related

text: bands

text: extend

text: to

text: the

text: Fig.

text: shows

text: the

text: surface

text: FSs

text: and

text: energy

text: dispersion

text: higher

text: energy

text: region

text: above

text: the

text: Weyl

text: points,

text: as

text: indicated

text: corresponding

text: to

text: the

text: two

text: diﬀerently

text: terminated

text: surfaces.

text: in

text: Fig.

text: Therefore,

text: to

text: observe

text: the

text: Fermi

text: arcs

text: from

text: Fixing

text: the

text: energy

text: at

text: the

text: Weyl

text: points

text: for

text: the

text: Sn-terminated

text: the

text: S-terminal

text: by

text: ARPES,

text: an

text: electron-doped

text: sample

text: and

text: surface,

text: one

text: can

text: easily

text: ﬁnd

text: the

text: single

text: surface

text: FS

text: starting

text: surface

text: electron-doping

text: is

text: necessary.

text: However,

text: we

text: note

text: from

text: one

text: Weyl

text: point

text: to

text: another

text: Weyl

text: point

text: with

text: the

text: that

text: although

text: potassium

text: doping

text: can

text: be

text: used

text: to

text: achieve

text: opposite

text: chirality,

text: which

text: is

text: the

text: typical

text: expected

text: behavior

text: small

text: upward

text: shifts

text: in

text: the

text: surface

text: chemical

text: of

text: Fermi

text: arcs.

text: Taking

text: the

text: C

text: rotation

text: symmetry

text: into

text: this

text: method

text: does

text: not

text: work

text: well

text: in

text: every

text: situation.

text: account,

text: there

text: are

text: three

text: such

text: Fermi

text: arcs

text: in

text: the

text: ﬁrst

text: BZ,

text: as

text: indicated

text: in

text: Fig.

text: d).

text: Although

text: a

text: little

text: part

text: of

text: Moreover,

text: in

text: a

text: signiﬁcant

text: diﬀerence

text: to

text: WSMs

text: that

text: the

text: Fermi

text: arc

text: merging

text: into

text: the

text: bulk,

text: the

text: Fermi

text: arc

text: still

text: exhibit

text: time-reversal

text: symmetry,

text: the

text: surface

text: state

text: measurements

text: extends

text: to

text: about

text: of

text: the

text: reciprocal

text: lattice

text: vector,

text: of

text: Co

text: Sn

text: S

text: need

text: to

text: consider

text: the

text: eﬀects

text: of

text: magnetic

text: which

text: is

text: suﬃciently

text: long

text: to

text: be

text: detected

text: by

text: an

text: ARPES

text: domains.

text: The

text: size

text: of

text: these

text: domains

text: are

text: typically

text: around

text: ##LTFigure##


text: 28 KRZYSZTOF BURDZY AND TVRTKO TADIC´ (cid:18) (cid:19) (cid:18) (cid:19) (cid:88) (cid:88) Proof. We use Lemma 6.10 (b) and (6.7) to see that ∞ ∞ P Xρα(n−1) ≤ Λ(1 + ε) = (cid:88)P X(cid:18) − − ≤(cid:19)Λ(1 + ε) ρα(n) log nα ρα(n) ρα(n 1) log nα n=1 n=1 ∞ ≥ P ≤ Λ(1 + ε) S . log nα (cid:18) (cid:19) (cid:18) (cid:19) n=1 The random variable S is IED(Λ) by Theorem 6.4, so we can use Lemma 2.7 to write P ≤ Λ(1 + ε) ≥ P ≤ Λ(1 + δ) ≥ − S S c n α/(1+δ). log nα (cid:18) log nα/(1+(cid:19)δ) δ (cid:88) It follows that ∞ P − ≤ Λ(1 + ε) ∞ Xρα(n 1) = . ρα(n) log nα n=2 − ≥ This, the fact that the random variables Xρα(n 1), n 2, are jointly independent and ρα(n) (cid:3) the Borel-Cantelli lemma imply the claim made in the lemma. (cid:16) (cid:17) → ∞ Lemma 6.12. When n , − − → log ρ (n) X Xρα(n 1) 0 a.s. α ρα(n) ρα(n)   (cid:89) Proof. We ﬁrst use (6.9) and then (6.8) with m = 0 to obtain   − − ρα(n) X Xρα(n 1) =  A  X −    ρα(n) ρα(n) (cid:89)− k ρα(cid:88)(n 1) (cid:89) k=ρα(n 1)+1      − − ρα(n) ρα(n 1) ρα(n 1)   = A A B k k j (cid:88) (cid:89) − k=ρα(n 1)+1  j=1 k=j+1 − ρα(n 1) ρα(n) = A B . k j j=1 k=j+1   We apply assumption (ii) of Theorem 6.7, a part of assumption (i) (namely, that A ’s − − k (cid:88) (cid:89) (cid:88) and B ’s are nonnegative), and the change of index i = ρ (n 1) j, to see that k   α − − − − ρα(n 1) ρα(n) ≤ ρα(n 1) − X Xρα(n 1) = A B βρα(n) jB ρα(n) ρα(n) (cid:88) k j j j=1 k=j+1 j=1 − − ρα(n 1) 1 − − = βρα(n) ρα(n 1)+iB − − . ρα(n 1) i i=0 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: KRZYSZTOF

text: BURDZY

text: AND

text: TVRTKO

text: TADIC´

text: Proof.

text: We

text: use

text: Lemma

text: (b)

text: and

text: to

text: see

text: that

text: P

text: ε)

text: ε)

text: ρα(n)

text: log

text: nα

text: ρα(n)

text: ρα(n

text: log

text: nα

text: P

text: ε)

text: S

text: log

text: nα

text: The

text: random

text: variable

text: S

text: is

text: IED(Λ)

text: by

text: Theorem

text: so

text: we

text: can

text: use

text: Lemma

text: to

text: write

text: P

text: ε)

text: P

text: δ)

text: S

text: S

text: c

text: n

text: log

text: nα

text: log

text: δ

text: It

text: follows

text: that

text: P

text: ε)

text: Xρα(n

text: ρα(n)

text: log

text: nα

text: This,

text: the

text: fact

text: that

text: the

text: random

text: variables

text: Xρα(n

text: n

text: are

text: jointly

text: independent

text: and

text: ρα(n)

text: the

text: Borel-Cantelli

text: lemma

text: imply

text: the

text: claim

text: made

text: in

text: the

text: lemma.

text: Lemma

text: When

text: n

text: log

text: ρ

text: (n)

text: X

text: Xρα(n

text: a.s.

text: α

text: ρα(n)

text: ρα(n)

text: Proof.

text: We

text: ﬁrst

text: use

text: and

text: then

text: with

text: m

text: to

text: obtain

text: ρα(n)

text: X

text: Xρα(n

text: A

text: X

text: ρα(n)

text: ρα(n)

text: k

text: k=ρα(n

text: ρα(n)

text: ρα(n

text: ρα(n

text: A

text: A

text: B

text: k

text: k

text: j

text: k=ρα(n

text: ρα(n

text: ρα(n)

text: A

text: B

text: k

text: j

text: We

text: apply

text: assumption

text: (ii)

text: of

text: Theorem

text: a

text: part

text: of

text: assumption

text: (i)

text: (namely,

text: that

text: A

text: ’s

text: k

text: and

text: B

text: ’s

text: are

text: nonnegative),

text: and

text: the

text: change

text: of

text: index

text: i

text: ρ

text: (n

text: j,

text: to

text: see

text: that

text: k

text: α

text: ρα(n

text: ρα(n)

text: ρα(n

text: X

text: Xρα(n

text: A

text: B

text: βρα(n)

text: jB

text: ρα(n)

text: ρα(n)

text: k

text: j

text: j

text: ρα(n

text: βρα(n)

text: ρα(n

text: ρα(n

text: i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 32 KRZYSZTOF BURDZY AND TVRTKO TADIC´ ∞ Proof. (a) We integrate(cid:18)the density of(cid:19)(A, B) w(cid:18)ith respect to (cid:19)b over (0, ) to compute the density of A. (cid:90) −(a1/2−1)2+1 − −(a1/2+1)2+1 ∞ exp 2 4 exp 2 4 P ∈  (cid:18)b √ (cid:19) b (cid:18) (cid:19)  (A da) = dbda (cid:90) (cid:90)  2π ab2  0    −(a1/2−1)2+1 −(a1/2+1)2+1  ∞ exp 2 4 ∞ exp 2 4 1√ b − b = (cid:34) db (cid:35) db da 2π a b2 b2 0 0 (cid:0) (cid:1) (cid:0) (cid:1) 1√ 1 − 1 = − da 2π(cid:0) a a(cid:1)1/2 1 2 + 1 a1/2 + 1 2 + 1 2 4 2 4 da = . π a2 + 1 4 √ (b) In the following calculation, we use the substitution u = a. (cid:18) (cid:19) (cid:18) (cid:19) P ∈ (B(cid:90) db) −(a1/2−1)2 − −(a1/2+1)2 ∞ exp 2 exp 2 − (cid:18) b (cid:19) √ (cid:18) b(cid:19) e√ 1/(4b) = da db (cid:90) 2 πab πb3/2 0 −(u−1)2 − −(u+1)2 ∞ exp 2 exp 2 −  (cid:18) b √(cid:19) b (cid:18) e√ 1/((cid:19)4b)  = du db (cid:90) (cid:90)  πb πb3/2  0    −(u−1)2 −(u+1)2  ∞ exp 2 ∞ exp 2 − √ b − √ b e√ 1/(4b) = (cid:32) du (cid:33) du db (cid:90) πb (cid:90) πb πb3/2 0 0 ∞ − ∞ − − (cid:32) e√u2/b (cid:33)− e√u2/b e√ 1/(4b) (cid:90) = du du db −1 πb 1 πb πb3/2 2 2 − − (cid:32)(cid:90) 12 e√u2/b (cid:33)e√ 1/(4b) = du db −1 πb πb3/2 2 √1 e√−v2/2 e√−1/(4b) 2b = dv db. −√1 2π πb3/2 2b (cid:90) (cid:112) This proves (7.3). We use (7.3) and the following facts, √1 e√−v2/2 √1 − lim b/2 2b dv = and lim e 1/(4b) = 1, →∞ →∞ b −√1 2π 2π b 2b ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: KRZYSZTOF

text: BURDZY

text: AND

text: TVRTKO

text: TADIC´

text: Proof.

text: (a)

text: We

text: density

text: B)

text: respect

text: to

text: over

text: to

text: compute

text: the

text: density

text: of

text: A.

text: exp

text: exp

text: P

text: b

text: (A

text: da)

text: dbda

text: 2π

text: exp

text: exp

text: 1√

text: b

text: b

text: db

text: db

text: da

text: 2π

text: a

text: 1√

text: da

text: a

text: da

text: π

text: (b)

text: In

text: the

text: following

text: calculation,

text: we

text: use

text: the

text: substitution

text: u

text: a.

text: P

text: db)

text: exp

text: exp

text: b

text: e√

text: da

text: db

text: πab

text: exp

text: exp

text: b

text: b

text: e√

text: du

text: db

text: πb

text: exp

text: exp

text: b

text: b

text: e√

text: du

text: du

text: db

text: πb

text: πb

text: e√

text: du

text: du

text: db

text: −1

text: πb

text: πb

text: du

text: db

text: −1

text: πb

text: √1

text: dv

text: db.

text: −√1

text: 2π

text: This

text: proves

text: We

text: use

text: and

text: the

text: following

text: facts,

text: √1

text: √1

text: lim

text: dv

text: and

text: lim

text: e

text: b

text: −√1

text: 2π

text: 2π

text: b

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text: INVERSE EXPONENTIAL DECAY 33 (cid:90) → ∞ to conclude that, when x , ∞ P ∼ 1 − 1 (B > x) b 2db = . (cid:90) π πx x This proves (7.5). Since √1 e√−v2/2 2b lim dv = 1, b→0+ −√1 (cid:90) 2π → 2b we obtain for x 0+, − P ∼ x e√ 1/(4b) (B < x) db. πb3/2 0 This proves (7.4). The claim that B is an IED(1)-random variable follows from (7.4) 4 (cid:3) by the same arguments as in Lemma 2.5. Lemma 7.2. If (A , B ) and (A , B ) are independent random vectors with the density 1 1 2 2 (7.1) then P ≤ − 3 −1 (7.6) lim sup ε log (A B + B < ε) < . → 2 1 2 10 4 ε 0+ Remark 7.3. Suppose that (A , B ) and (A , B ) are i.i.d., and A and B are inde- 1 1 2 2 1 1 pendent with distributions (7.2) and (7.3). Then ess inf(A ) = 0 and B is IED(1). By P − 1 1 4 Corollary 6.3, lim → ε log (A B + B < ε) = 1. However, (7.6) shows that we do ε 0+ 2 1 2 4 not have the same conclusion when A and B are not independent. 1 1 (cid:18) (cid:18) (cid:19) (cid:19) Proof of Lemma 7.2. First, recall that by Lemma 2.7, for every δ > 0 there exists C > 0 such that for all x > 0, P ≤ − 1 − − (B < x) C exp δ x 1 . 1 4 The second inequality in the following calculation is justiﬁed by the above formula. Later in the calculation, we will(cid:18)use the substit(cid:19)ution y =(cid:18) a1/2. (cid:19) P (A(cid:90) B(cid:90)+ B < ε) 2 1 2 −(a1/2−1)2+1 − −(a1/2+1)2+1 ∞ exp 2 4 exp 2 4 ε P (cid:18) b (cid:19)√ b = (aB + b < ε) da db (cid:90) (cid:90) (cid:18) 1 (cid:19) 2πb2 a 0 0 −(a1/2−1)2+1 ∞ − exp 2 4 ≤ ε P ε b √b B < da db 1 a 2πb2 a 0 0 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: INVERSE

text: EXPONENTIAL

text: DECAY

text: to

text: conclude

text: that,

text: when

text: x

text: P

text: (B

text: x)

text: b

text: π

text: πx

text: x

text: This

text: proves

text: Since

text: √1

text: lim

text: dv

text: −√1

text: 2π

text: we

text: obtain

text: for

text: x

text: P

text: x

text: e√

text: (B

text: x)

text: db.

text: This

text: proves

text: The

text: claim

text: that

text: B

text: is

text: an

text: variable

text: follows

text: from

text: by

text: the

text: same

text: arguments

text: as

text: in

text: Lemma

text: Lemma

text: If

text: (A

text: B

text: and

text: (A

text: B

text: are

text: independent

text: random

text: vectors

text: with

text: the

text: density

text: then

text: P

text: −1

text: lim

text: sup

text: ε

text: log

text: (A

text: B

text: B

text: ε)

text: ε

text: Remark

text: Suppose

text: that

text: (A

text: B

text: and

text: (A

text: B

text: are

text: i.i.d.,

text: and

text: A

text: and

text: B

text: are

text: inde-

text: pendent

text: with

text: distributions

text: and

text: Then

text: ess

text: inf(A

text: and

text: B

text: is

text: By

text: P

text: Corollary

text: lim

text: ε

text: log

text: (A

text: B

text: B

text: ε)

text: However,

text: shows

text: that

text: we

text: do

text: ε

text: not

text: have

text: the

text: same

text: conclusion

text: when

text: A

text: and

text: B

text: are

text: not

text: independent.

text: Proof

text: of

text: Lemma

text: First,

text: recall

text: that

text: by

text: Lemma

text: for

text: every

text: δ

text: there

text: exists

text: C

text: such

text: that

text: for

text: all

text: x

text: P

text: (B

text: x)

text: C

text: exp

text: δ

text: x

text: The

text: second

text: inequality

text: in

text: the

text: following

text: calculation

text: is

text: justiﬁed

text: by

text: the

text: above

text: formula.

text: Later

text: in

text: the

text: calculation,

text: we

text: the

text: y

text: P

text: B

text: ε)

text: exp

text: exp

text: ε

text: P

text: b

text: b

text: (aB

text: b

text: ε)

text: da

text: db

text: a

text: exp

text: ε

text: P

text: ε

text: b

text: √b

text: B

text: da

text: db

text: a

text: a

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text: INVERSE EXPONENTIAL DECAY 7 Proof of Theorem 2.8. If X is an IED-random(cid:18) variable (cid:19)with the parameter λ , then by 1 1 Lemma 2.7, − ≤ P P x ≤ − − c e α1(λ1+δ)/x (α X < x) = X < C e α1λ1(1 δ)/x, δ 1 1 1 δ α 1 for all x > 0. It follows that − ≤ P ≤ P ≤ − − α (λ + δ) lim inf x log (α X < x) lim sup x log (α X < x) α λ (1 δ). 1 1 x→0+ 1 x→0+ 1 1 1 → By letting δ 0 we see that α X is an IE(cid:0)D-random va(cid:1)riable with the parameter α λ . 1 1 1 1 By the argument given above, if n = 2, √it is su√ﬃcient to show that X + X is an 1 2 ≥ 2 IED-random variable with the parameter λ + λ . Then the general case n 2 1 2 follows by induction. First we deal with the case λ + λ > 0 and assume without loss of generality that 1 2 λ > 0. (cid:90) 1 It follows from independence of X and X that 1 2 P x P − P ∈ (X + X < x) = (X < x y) (X dy). 1 2 1 2 0 In the following calculation we use (in this order) the last formula, the upper bound in (2.4), integration by parts, the upper bound in (2.4) again, and Lemma 2.9 (i). For ∈ ∞ any δ, ε (0, 1) and some c(cid:90)onstants C1, C2 < , δ δ (2.6) P(X(cid:90)1 + X2 < x) ≤ x Cδ1e−λ1x(−1−yδ)(cid:16)P(X2 ∈ dy(cid:17)) 0 P − − = (cid:90) x C1λ (1 − δ) (X2 <(cid:16)y)(cid:16)e−xp λ1x(−1 yδ) dy(cid:17)(cid:17) δ 1 (x y)2 0 − − − ≤ x − exp (cid:16) (cid:16)λ1(−1 δ) + λ2(1 δ) (cid:17)(cid:17) (cid:16) (cid:16) (cid:17)(cid:17) (2.7) (cid:90) C1C2λ (1 δ) x −y y dy δ δ 1 (x y)2 0 − − − − − − x − exp (cid:16)(cid:112)λ1(1x−δ)y(1 ε) + λ2(1y δ) (cid:112)exp (cid:17)λ1(x1−yδ)ε = C1C2λ (1 δ) − dy δ δ 1   (x y)2  0    − − − 2 λ (1 δ)(1 ε) + λ (1 δ) ≤ − − 1 2 × C1C2λ (1 (cid:16)δ) e(cid:16)xp (cid:17)(cid:17) δ (cid:90)δ 1 x − − × x exp λ1(1− δ)ε x y − dy (x y)2 0 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: INVERSE

text: EXPONENTIAL

text: DECAY

text: Proof

text: of

text: Theorem

text: If

text: X

text: is

text: an

text: variable

text: the

text: parameter

text: λ

text: then

text: by

text: Lemma

text: P

text: P

text: x

text: c

text: e

text: (α

text: X

text: x)

text: X

text: C

text: e

text: δ)/x,

text: δ

text: δ

text: α

text: for

text: all

text: x

text: It

text: follows

text: that

text: P

text: P

text: α

text: (λ

text: δ)

text: lim

text: inf

text: x

text: log

text: (α

text: X

text: x)

text: lim

text: sup

text: x

text: log

text: (α

text: X

text: x)

text: α

text: λ

text: δ).

text: By

text: letting

text: δ

text: we

text: see

text: that

text: α

text: X

text: is

text: an

text: with

text: the

text: parameter

text: α

text: λ

text: By

text: the

text: argument

text: given

text: above,

text: if

text: n

text: √it

text: is

text: su√ﬃcient

text: to

text: show

text: that

text: X

text: X

text: is

text: an

text: IED-random

text: variable

text: with

text: the

text: parameter

text: λ

text: λ

text: Then

text: the

text: general

text: case

text: n

text: follows

text: by

text: induction.

text: First

text: we

text: deal

text: with

text: the

text: case

text: λ

text: λ

text: and

text: assume

text: without

text: loss

text: of

text: generality

text: that

text: λ

text: It

text: follows

text: from

text: independence

text: of

text: X

text: and

text: X

text: that

text: P

text: x

text: P

text: P

text: (X

text: X

text: x)

text: (X

text: x

text: y)

text: (X

text: dy).

text: In

text: the

text: following

text: calculation

text: we

text: use

text: (in

text: this

text: order)

text: the

text: last

text: formula,

text: the

text: upper

text: bound

text: in

text: integration

text: by

text: parts,

text: the

text: upper

text: bound

text: in

text: again,

text: and

text: Lemma

text: (i).

text: For

text: any

text: δ,

text: ε

text: and

text: some

text: δ

text: δ

text: x)

text: x

text: P

text: x

text: δ)

text: yδ)

text: δ

text: (x

text: x

text: exp

text: δ)

text: δ)

text: δ)

text: x

text: −y

text: y

text: dy

text: δ

text: δ

text: (x

text: x

text: exp

text: ε)

text: δ)

text: δ)

text: dy

text: δ

text: δ

text: (x

text: λ

text: ε)

text: λ

text: δ)

text: δ

text: x

text: x

text: exp

text: δ)ε

text: x

text: y

text: dy

text: (x

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text: [9] Lamb J. S. W., Stenkin O. V. Newhouse regions for reversible systems with inﬁnitely many stable, unstable and elliptic periodic orbits // Non- linearity. 2004. Vol. 17. No. 4. pp. 1217. [10] Politi A., Oppo G. L., Badii R. Coexistence of conservative and dissi- pative behavior in reversible dynamical systems // Physical Review A. 1986. Vol. 33. No. 6. p. 4055. [11] Roberts J. A. G., Quispel G. R. W. Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems // Physics Reports. 1992. Vol. 216. No. 2-3. pp. 63–177. [12] Lamb J. S. W., Roberts J. A. G. Time-reversal symmetry in dynamical systems: a survey // Physica D: Nonlinear Phenomena. 1998. Vol. 112. No. 1. pp. 1–39. [13] Topaj D., Pikovsky A. Reversibility vs. synchronization in oscillator lattices // Physica D: Nonlinear Phenomena. 2002. Vol. 170. No. 2. pp. 118–130. [14] Gonchenko A.S., Gonchenko S.V., Kazakov A.O., Turaev D.V. On the phenomenon of mixed dynamics in PikovskyTopaj system of coupled rotators // Physica D: Nonlinear Phenomena. 2017. Vol. 350. pp. 45–57. [15] Gonchenko A. S., Gonchenko S. V., Kazakov A. O. Richness of chaotic dynamics in nonholonomic models of a Celtic stone // Regular and Chaotic Dynamics. 2013. Vol. 18. No. 5. pp. 521–538. [16] Kazakov A. O. Strange attractors and mixed dynamics in the problem of an unbalanced rubber ball rolling on a plane // Regular and Chaotic Dynamics. 2013. Vol. 18. No. 5. pp. 508–520. [17] Kazakov A. On chaotic dynamics in the Suslov problem // Dynamics, bifurcations and chaos 2015 (DBC II): Ext. Abstr. Int. Conf.Sch., Nizhni Novgorod, July 2024, 2015. Nizhni Novgorod: Lobachevsky State Univ., 2015. P. 2130. [18] Bizyaev I. A., Borisov A. V., Kazakov A. O. Dynamics of the suslov prob- lem in a gravitational ﬁeld: Reversal and strange attractors // Regular and Chaotic Dynamics. 2015. Vol. 20. No. 5. pp. 605–626. 14

label: reference

text: Lamb

text: J.

text: S.

text: W.,

text: Stenkin

text: O.

text: V.

text: Newhouse

text: regions

text: for

text: reversible

text: systems

text: with

text: inﬁnitely

text: many

text: stable,

text: unstable

text: and

text: elliptic

text: periodic

text: orbits

text: Non-

text: linearity.

text: Vol.

text: No.

text: pp.

text: Politi

text: A.,

text: Oppo

text: G.

text: L.,

text: Badii

text: R.

text: Coexistence

text: of

text: conservative

text: and

text: dissi-

text: pative

text: behavior

text: in

text: reversible

text: dynamical

text: systems

text: Physical

text: Review

text: A.

text: Vol.

text: No.

text: p.

text: Roberts

text: J.

text: A.

text: G.,

text: Quispel

text: G.

text: R.

text: W.

text: Chaos

text: and

text: time-reversal

text: symmetry.

text: Order

text: and

text: chaos

text: in

text: reversible

text: dynamical

text: systems

text: Physics

text: Reports.

text: Vol.

text: No.

text: pp.

text: Lamb

text: J.

text: S.

text: W.,

text: Roberts

text: J.

text: A.

text: G.

text: Time-reversal

text: symmetry

text: in

text: dynamical

text: systems:

text: a

text: survey

text: Physica

text: D:

text: Nonlinear

text: Phenomena.

text: Vol.

text: No.

text: pp.

text: Topaj

text: D.,

text: Pikovsky

text: A.

text: Reversibility

text: vs.

text: synchronization

text: in

text: oscillator

text: lattices

text: Physica

text: D:

text: Nonlinear

text: Phenomena.

text: Vol.

text: No.

text: pp.

text: Gonchenko

text: A.S.,

text: Gonchenko

text: S.V.,

text: Kazakov

text: A.O.,

text: Turaev

text: D.V.

text: On

text: the

text: phenomenon

text: of

text: mixed

text: dynamics

text: in

text: PikovskyTopaj

text: system

text: of

text: coupled

text: rotators

text: Physica

text: D:

text: Nonlinear

text: Phenomena.

text: Vol.

text: pp.

text: Gonchenko

text: A.

text: S.,

text: Gonchenko

text: S.

text: V.,

text: Kazakov

text: A.

text: O.

text: Richness

text: of

text: chaotic

text: dynamics

text: in

text: nonholonomic

text: models

text: of

text: a

text: Celtic

text: stone

text: Regular

text: and

text: Chaotic

text: Dynamics.

text: Vol.

text: No.

text: pp.

text: Kazakov

text: A.

text: O.

text: Strange

text: attractors

text: and

text: mixed

text: dynamics

text: in

text: the

text: problem

text: of

text: an

text: unbalanced

text: rubber

text: ball

text: rolling

text: on

text: a

text: plane

text: Regular

text: and

text: Chaotic

text: Dynamics.

text: Vol.

text: No.

text: pp.

text: Kazakov

text: A.

text: On

text: chaotic

text: dynamics

text: in

text: the

text: Suslov

text: problem

text: Dynamics,

text: bifurcations

text: and

text: chaos

text: (DBC

text: II):

text: Ext.

text: Abstr.

text: Int.

text: Conf.Sch.,

text: Nizhni

text: Novgorod,

text: July

text: Nizhni

text: Novgorod:

text: Lobachevsky

text: State

text: Univ.,

text: P.

text: Bizyaev

text: I.

text: A.,

text: Borisov

text: A.

text: V.,

text: Kazakov

text: A.

text: O.

text: Dynamics

text: of

text: the

text: suslov

text: prob-

text: lem

text: in

text: a

text: gravitational

text: ﬁeld:

text: Reversal

text: and

text: strange

text: attractors

text: Regular

text: and

text: Chaotic

text: Dynamics.

text: Vol.

text: No.

text: pp.