Eigenvalue detachment BBP transition and constrained Brownian motion Alexander Gorsky1 Sergei Nechaev2 and Alexander Valov3

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Eigenvalue detachment, BBP transition and constrained

Brownian motion

Alexander Gorsky1, Sergei Nechaev2, and Alexander Valov3

1Institute for Information Transmission Problems RAS, 127051 Moscow, Russia

2LPTMS, CNRS – Universit´e Paris Saclay, 91405 Orsay Cedex, France

3N.N. Semenov Federal Research Center for Chemical Physics RAS, 119991 Moscow, Russia

(Dated: today)

We discuss the eigenvalue detachment transition in terms of scaling of ﬂuctuations

in ensembles of paths located near convex boundaries of various physical nature.

We consider numerically the BBP-like (Baik-Ben Arous-P´ech´e) transition from the

Gaussian to the Tracy-Widom scaling of ﬂuctuations in several statistical systems

for both canonical and microcanonical ensembles and identify the corresponding

control parameter in each case. In particular, for ﬁxed path length (microcanonical)

ensemble of paths located in the vicinity of a partially permeable semicircle, the

transition occurs at the critical value of a permeability. The Tracy-Widom regime

and the BBP-like transition for ﬂuctuations are discussed in terms of the Jakiw-

Teitelbom (JT) gravity with a radial cutoﬀ which, in turn, has an interpretation

as a ensemble of ﬁxed length world-line geometrically constrained trajectories of a

charged particle in an eﬀective transversal magnetic ﬁeld.

I. INTRODUCTION: GENERAL FRAMEWORKS OF UNCONVENTIONAL

TRANSITIONS

Scaling laws of ﬂuctuations provide the convenient instrument of probing statistical be-

havior of the system. Two conventional laws, “Gaussian” and “Tracy-Widom” manifest

themselves in a huge variety of statistical systems. The common Gaussian law is typically

related with the central limit theorem for a large number of independent random variables,

while the Tracy-Widom (TW) law [1] emerges in extreme statistics of a large number of cor-

related random variables. The behavior of some observables in scaling regimes is governed

by universal solutions of diﬀerential equations. For example, the solution to the Painleve II

yields the scaling function of the TW regime. It has been explicitly shown in [2] that the

eigenvalue detachment can be microscopically identiﬁed with the transition for the largest

eigenvalue, λmax, from the TW distribution (when λmax correlates with the bulk) to the

Gaussian one (when the correlations of λmax with the bulk are absent). The new “interme-

diate” distribution emerges exactly at the transition point.

The detachment of the largest eigenvalue from the bulk of the spectrum is the phenomena

arXiv:2210.01201v1 [cond-mat.stat-mech] 3 Oct 2022

which can be seen in various physical situations. Historically such a detachment has been

ﬁrst discussed in detail in the context of spin glasses [3]. In this case the control parameter

was identiﬁed with the eigenvalue of the matrix which centers the Gaussian ensemble with

the shifted mean. More recently, this phenomenon was considered in the probability theory

for the behavior of eigenvalues of some covariance matrix [2]. The control parameter is the

deviation of a single eigenvalue of the covariance matrix from the unity. Later more general

situation with several control parameters was discussed as well. Exactly at the transition

point the new universality class has been identiﬁed which distribution which now is known

as the “Baik-Ben Arous-P´ech´e” (BBP) distribution. Similar to the Tracy-Widom (TW)

distribution which is given by the solution to the Painleve II equation with the speciﬁc

asymptotics, the BBP distribution is described by the pair of diﬀerential equations which

involve the solution to Painleve II equation with the peculiar monodromy [4].

The shifted mean Gaussian ensemble has been generalized to the chiral case in [5] and the

position of the separated eigenvalue as a function of the order parameter has been derived

analytically. Later on more eigenvalue detachment phase transitions were found in such

physical problems as: last passage percolation [6]; Asymmetric Simple Exclusion Process

(ASEP) with particular spiked initial conditions [7, 8]; q-version of Totally Asymmetric

Simple Exclusion Process (q-TASEP) with slow particles [9]; percolation in 2D [10]; spin

glass-paramagnetic transition in the mean ﬁeld approximation [11]. In all these cases the

nature of the phenomena is one and the same: ﬁrst one zooms the spectral edge where the

TW distribution emerges, and then the perturbation is introduced.

In our study we focus our attention not at the distribution of spectral ﬂuctuations, rather

at the distribution of paths’ ﬂuctuations in space-time. The fermionic nature of eigenvalues

in the matrix model can be mapped onto the vicious walkers problem. So, the detachment of

one eigenvalue from the bulk of the spectrum gets mapped onto the emergence of the outliers

in the Brownian ensemble. The dynamics of the Brownian motion of large number of walkers

with a few outliers has been also discussed in the mathematical literature in [12] where the

transition from the Gaussian to TW ﬂuctuations has been derived analytically. Instead

of looking at detached eigenvalues, the same pattern can be modelled in the ensemble of

individual random walkers nearby the extended defect. In the polymer language, the BBP

transition occurs when a (1+1)D random walk is located exactly at the threshold of the

formation of a bound state with some type of defect, or attractive boundary. As shown in

examples below, the interaction of a random path with a defect can be introduced either

explicitly of eﬀectively.

The example of an explicit interaction is provided in the work [13], where the BBP-

like transition emerges when a part of a polymer trajectory gets localized on an extended

defect at some critical value of coupling between the polymer and the defect. Mapping

of the polymer problem onto the matrix model in the context of the BBP transition has

been discussed in [13]. Conceptually the model under consideration is as following. Take

a random matrix playing a role of a transfer-matrix for some lattice model and suppose

that the spectrum of this matrix shares the Wigner semicircle law (that is a rather generic

supposition). It is known that at the edge of the semicircle the largest eigenvalue has the

Tracy-Widom distribution. Let one deform the transfer-matrix by a perturbation involving

some coupling constant, u. Above a critical coupling, uc, one eigenvalue detaches from the

continuum, which is the “spectral manifestation” of the BBP transition.

The eﬀective interaction of a path with an extended defect has been studied in [14–17]

in the context of the Ferrari-Spohn problem [18]. Being rephrased in polymer terms, the

problem is as follows: the part of a polymer trajectory located in the vicinity of a convex

void boundary experiences the transition between diﬀerent ﬂuctuation regimes as a function

of a boundary curvature. The corresponding behavior has been interpreted in [19, 20] as a

“shadowing” a path by a convex impermeable boundary on which the path is leaning. In

such a setup the key point is the consideration of the microcanonical ensemble of ﬂuctuating

paths of a ﬁxed length, N. The condition N=cR is imposed on paths with ends ﬁxed at

opposite extremities of a diameter of a convex void (a semicircle of radius R). Varying c,

which is the control parameter, one sees the transition from TW scaling to the Gaussian

one, which we interpret as a BBP transition. The transition here is induced by ﬁxing the

path’s length which forces the trajectories to nestle against the disc boundary and cis the

control parameter which governs the strength of the “pinching force” and plays a role of an

eﬀective curvature of a disc’s boundary for a path of a given length.

In the present work we discuss numerically manifestations of the BBP transition in various

scenarios for a single random path nearby the convex boundary. Speciﬁcally, we consider

several formulations:

(i) Microcanonical ensemble of random paths of ﬁxed length, L=cR, above an imper-

meable disc of radius Rwhere cis the control parameter for BBP transition

(ii) Microcanonical ensemble of random paths of ﬁxed length L=cR, above a partially

impermeable disc of radius R, in which the fraction of chain monomers inside the disc

is controlled by the parameter η. We see that ηis the control parameter for the BBP

transition;

(iii) Canonical ensemble of random paths whose length, L, is controlled by the chemical

potential sabove the ﬁsh-like defect. The angle at the cusp is the control parameter.

The interplay between the BBP transition in terms of spectral and spatial ﬂuctuations

can be naturally understood in the holographic framework. The radial coordinate in the

hyperbolic plane has the meaning of the energy scale in the boundary theory. Hence indeed

the BBP transition in the scaling regime of ﬂuctuations for eigenvalues in the boundary

theory ﬁts with the radial ﬂuctuations in its holographic dual. Since the BBP transition

concerns the maximal eigenvalue, the introduction of some radial cutoﬀ is expected. We

shall comment on the TW regime and the BBP transition in Jakiw-Teitelbom (JT) gravity

holographically described via the quantum mechanics with large number degrees of freedom.

The key point is the identiﬁcation of the partition function of JT gravity as the function of

the ensemble of the ﬁxed length paths of a charged particle in the hyperbolic plane in the

transversal magnetic ﬁeld [21, 22]. To get the TW regime we should introduce the radial

cutoﬀ Rcut which corresponds the energy cutoﬀ in the boundary theory and then tune the

temperature is such way that the cutoﬀ radius approaches the Larmour radius RLar of a

charged particle in the transversal magnetic ﬁeld. In this case we arrive at the framework

which is similar to one discussed for the bunch of trajectories nearby the hard circle (which

mimics the cutoﬀ radius). The quotient Rcut/RLar is the control parameter in this case.

The paper is structured as follows. In Section II we recall the general framework of

the BBP phase transition. In Section III we consider diﬀerent formulations of the model

and observe numerically the transition from KPZ to Gaussian regime for ﬂuctuations in each

case. In Section IV we formulate the Tracy-Widon regime and the BBP-like transition in the

JT gravity with the radial cutoﬀ under particular limitation of parameters. In Section IV we

summarize obtained results and speculate about possible further developments. In Appendix

A we consider the trajectory in the background of an attractive defect. Such a scenario has

been discussed for the ﬂat defect in [13] while we have presented another example of a similar

kind for the convex defect. Considered setting describes the induced false vacuum decay in

the (1+1)-dimensional space-time. In Appendix B we use the nonlinear UMAP method of

the dimensional data reduction to relate the BBP transition with the changes of the data

spot in the abstract two-dimensional plane. In Appendix C we present for completeness

some known formulae concerning the BBP transition.

II. TRACY-WIDOM SCALING LAW FOR FLUCTUATIONS AND BAIK-BEN

AROUS-PECH´

E TRANSITION

Here we brieﬂy recall the origin of the Tracy-Widom (TW) distribution and the formula-

tion of the Baik-Ben Arous-P´ech´e (BBP) transition for ﬂuctuational statistics. Historically

the Tracy-Widom law has been identiﬁed for the Airy kernel in some random matrix models

A(x, y) = Z∞

Ai(z+x)Ai(z+y)dz =Ai0(x)Ai(y)−Ai0(y)Ai(x)

x−y(1)

The TW distribution, F0, can be expressed in terms of the Fredholm determinant

F0(s) = det 1−As(x, y)(2)

where As(x, y) is the corresponding kernel operator. The TW distribution can be expressed

in terms of solutions of the Painleve II equation as follows

F0(s) = exp −Z∞

x=s

(s−x)2u2(x)dx(3)

where function u(x) obeys the Painleve II equation

u00(x) = 2u3(x) + xu(x) (4)

subject to the speciﬁc boundary condition deﬁned by the asymptotics

u(x)∝ −Ai(x), x →+∞(5)

The function F0(s) describes the ﬂuctuations of the largest eigenvalue λmax in ensembles of

interacting particles. For instance for the Laguerre unitary ensemble one has

Pλmax −cN2/3≤s→F0(s) (6)

where cis parameter of the model.

It is known that the Tracy-Widom scaling is also provided by the solutions to the one-

dimensional KPZ equation for the height function h(x, t) developing in time t:

∂h(x, t)

∂t =1

2∂h(x, t)

∂x 2

∂2h(x, t)

∂x2+W(x, t) (7)

where W(x, t) is the white noise in (1+1)D space-time. Upon the Cole-Hopf transform

Z=eh(x,t), Eq.(7) it can be brought into the form

∂Z(x, t)

∂t =∂2Z(x, t)

∂x2+W(x, t)Z(x, t) (8)

where the function Z(x, t) can be interpreted as the partition function of the polymer in the

plane in an external random potential. At large tthe height function behaves as

h(0, t)∝ − t

24 +t

21/3

η(9)

where ηfor the wedge initial condition is the random Tracy-Widom distributed variable

coinciding with the distribution of the largest eigenvalue of the Gaussian Unitary Matrix

Ensemble (GUE) at large matrix sizes, N. The distribution is non-universal for KPZ and

depends on the initial condition in a KPZ equation.

In what follows we shall be interested in the interpretation of Z(x, t) as of the partition

function of N-step directed random walks (polymers) in a random media with the initial

condition Z(x, 0) = δ(x) and ending point located at (x, t). The large-Nbehavior of the free

energy of ensemble of such polymers can be derived from KPZ equation and reads [23, 24]:

F=Tln Z(0, N)∝ −NE0+aN 1

3(10)

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Eigenvaluedetachment,BBPtransitionandconstrainedBrownianmotionAlexanderGorsky1,SergeiNechaev2,andAlexanderValov31InstituteforInformationTransmissionProblemsRAS,127051Moscow,Russia2LPTMS,CNRS{UniversiteParisSaclay,91405OrsayCedex,France3N.N.SemenovFederalResearchCenterforChemicalPhysicsRAS,119991Mos...

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Eigenvalue detachment BBP transition and constrained Brownian motion Alexander Gorsky1 Sergei Nechaev2 and Alexander Valov3

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