Fluctuations of ring polymers

1.8 MB·English

by  Shlomi Medalion

Fluctuations of ring polymers
Shlomi Medalion,1 Erez Aghion,2 Hagai Meirovitch,2 Eli Barkai,1 and David A. Kessler2
1Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan,52900, Israel
2Department of Physics, Bar-Ilan University, Ramat-Gan, 52900, Israel
(Dated: January 27, 2015)
We present an exact solution for the distribution of sample averaged monomer to monomer dis-
tance of ring polymers. For non-interacting and weakly-interacting models these distributions cor-
respond to the distribution of the area under the reflected Bessel bridge and the Bessel excursion
respectively, and are shown to be identical in dimension d≥2. A symmetry of the problem reveals
thatdimensiondand4−dareequivalent,thusthecelebratedAirydistributiondescribingtheareal
distribution of the d = 1 Brownian excursion describes also a polymer in three dimensions. For a
5 self-avoidingpolymerindimensiondwefindnumericallythatthefluctuationsofthescaledaveraged
1 distance are nearly identical in dimension d = 2,3 and are well described to a first approximation
0 by the non-interacting excursion model in dimension 5.
2
n PACSnumbers: 05.40.Jc,02.50.-r,02.50.Ey,05.40.-a,05.10.Gg,05.20.Gg,36.20.Ey,36.20.Hb
a
J
The statistical mechanics of polymers has been well equation, whichdependsonboththedimensionalityand
5
2 studiedformanyyearsduebothtothenumerouspracti- theinteraction,andinturnyieldsaselectionruleforthe
cal applications of polymers as well as their many inter- solution. The resulting distributions are then compared

] esting properties. One signal finding is that the overall numerically to those measured in simulations for a ring
t
of sizeofapolymeroflengthN scalesasNνp,whereνp isa polymer with full excluded volume constraints.
dimension dependent critical exponent. This is reflected
s Aswehavenoted,constrainedrandomwalkslieatthe
. in the behavior of various observables, such as the aver-
t heart of our analysis. For rings, the primary constraint
a age end-to-end distance, and the radius of gyration (R )
m g is that the path returns to the origin after N steps.
[1, 2]. The scaling exponent is known to be sensitive to
Statistics of such constrained one dimensional Brownian
- the excluded volume interaction applied to part or all of
d paths have been the subject of much mathematical
the monomers. Other geometrical constraints applied to
n research [6–9]. These constrained paths have been given
o the chain, such as cyclization (where the first monomer various names, depending on the additional constraints
c is connected to the last one), leading to a “ring” poly-
imposed. The basic case is that of a Brownian bridge
[
mer, affect only the prefactor for these quantities. Ring where the return to the origin is the sole constraint. For

1 polymers are commonly found in many biological sys- a Brownian excursion, the path is also forbidden from
v tems e.g., bacterial and mitochondrial genomes, as well
reaching the origin in between. Majumdar and Comtet
1
asDNAplasmidsusedinmanymolecularbiologyexper-
used Brownian excursions to determine statistical prop-
5
iments [3]. Recently, ring polymers were also studied in
1 erties of the fluctuating Edwards Wilkinson interface in
6 the context of a model for chromosome territories in the an interval [10, 11]. The focus of most previous work
0 nucleus of eukaryotic cells [4]. has been on the constrained one dimensional Brownian
.
1 pathswhichdescribesinherentlynon-interactingsystems
Theconformationalfluctuationsofsomepolymermod-
0 (note that the problems of non-intersecting Brownian
els can be analyzed using the theory of random walks
5
excursions [12] or vicious random walkers [13] in higher
1 (RW)[1,2]. Inparticular,thefluctuationsofthepolymer
dimensions are exceptions). For the case of the fluctua-
: sizeareofphysicalandbiologicalimportance. Inthecur-
v tions of ring polymers, we need to extend the theory of
rent paper we study the distribution of sample-averaged
i
X monomer-to-monomer distance of ring polymers, both Brownian excursions and bridges to other dimensions.
Weaddresstheinfluenceofdifferentkindsofinteractions
r for ideal, noninteracting polymers as well as for poly-
a on the polymer structure, both analytically (for a single
mers with excluded volume. In particular, we exploit
point interaction) and numerically (for a polymer with
recent mathematical development on d dimensional con-
excluded volume interactions). These models yield rich
strained Brownian motion (defined below) [5] to find an
physical behaviors and open new questions.
exact expression for the distribution of sample averaged
monomer to monomer distance for both ideal ring poly-
mersandthosewithanadditionalexcludedvolumeinter- Polymer Models. We consider three lattice models of
action applied at a single point. This observable yields ring polymers with N bonds, each of length b, in d di-
insight on the sample averaged fluctuations of polymer mensions. Thesimplestpolymermodelisan“idealring”
sizes. An important ingredient of this calculation is the -aclosedchainwithoutexcludedvolume,wheredifferent
identification of the appropriate boundary conditions for monomers can occupy the same lattice site. While such
the underlying equation, a variant of the Feynman-Kac a polymer does not exist in nature, its global behavior is
2
thesameasthatofapolymerattheFloryθ-temperature
[1]. An ideal ring chain corresponds exactly to an unbi-
ased RW in d dimensions which starts and ends at the
origin, i.e. a d-dimensional bridge. In the second model,
the “weakly interacting ring polymer”, the first and last d=1 WI
d=2 IR
monomersaretiedtotheoriginandnoothermonomeris 2
d=2 WI
allowedtooccupythislatticesite. Thiscaseisequivalent
d=3 IR
to that of d-dimensional excursions. The third model is d=3 WI
1.5
a ring polymer with excluded volume interactions, also æ)A d=5 IR
d=5 WI
calledaself-avoidingwalk(SAW).Furtherdetailsonthe /Æ
A d=2 TH
polymer models and simulation methods are provided in (P 1 d=3 TH
the supplementary material (SM). We first consider the d=5 TH
ideal and weakly interacting ring models, for which we 0.5 d=2 SAW
d=3 SAW
can provide an analytic solution.
Bessel process. In the analogy between statistics of an
0
idealpolymerandaRW,thepositionoftheithmonomer, 1 2
r , corresponds to the position r of the random walker A / Æ Aæ
i i
after i time steps and N is proportional to the total ob-
servation time. The Bessel process [14, 15] describes the
dynamics of the distance r = |r| from the origin of a FIG.1. (coloronline)TheoreticalP+(A/(cid:104)A(cid:105))Eq. (10)indi-
Brownian particle in d dimensions. This process is de- mensiond=2,3,5nicelymatchessimulationsoftheidealring
scribed by the following Langevin equation: (IR) and the weakly interacting (WI) models, the exception
being the weakly interacting model in d = 2 due to critical
(d−1) slowing down (the simulations did not converge for N =106,
r˙ = +η(t), (1)
2r seetext). Thetheoreticalcurves(solidlines)forthetwomod-
els are identical for d ≥ 2. The d = 1 WI theory is identical
whereη(t)isGaussianwhitenoisesatisfying(cid:104)η(cid:105)=0and
tothed=3theory,whichthesimulationconfirms. TheSAW
(cid:104)η(t)η(t(cid:48))(cid:105)=δ(t−t(cid:48)). Onemaymapthepolymermodels
simulations in d=2,3 are practically indistinguishable from
to the Bessel process using (cid:104)r(t)2(cid:105) = dt = Nb2 = (cid:104)R2(cid:105), each other and from the theoretical curve corresponding to a
where R2 is the mean square end-to-end distance of an noninteracting ring in dimension d=5.
ideal linear polymer chain without constraints. In what
follows we choose b2 =d and t=N.
Bessel Excursions and Reflected Bridges. The pro-
A/N is the sample averaged distance of the monomers
cess r(t) with the additional constraint of starting and
from the origin. Specifically, let the area under the ran-
ending at the origin, is called a reflected (since r ≥ 0) (cid:82)t
domBesselcurver(t)bedenotedbyA = r(t)dt(the
Besselbridge. Thisprocesscorrespondstoanideal(non- B 0
subscript B is for Bessel). More generally the mapping
interacting) ring chain. Bessel excursions are paths still
of the processes implies that in the limit of large N the
described by Eq. (1) however with the additional con-
distribution of A /(cid:104)A (cid:105) is identical to the distribution
B B
straint that any path that reaches the origin (besides of A/(cid:104)A(cid:105) (or ¯l/(cid:104)¯l(cid:105)), with the corresponding constraints.
t=0 and t=N) is excluded. The Bessel excursion cor-
Numerical results. In Fig. 1 we plot the probability
respondstowhatwehavecalledthe“weaklyinteracting”
density function (PDF) of the scaled random variable
ring chain, where a multiple occupation of the origin is
A/(cid:104)A(cid:105). Both results of simulations and theory are pre-
not allowed. The mapping of the polymer models to the
sented, however at this stage let us focus on the main
Bessel process, allows us to extract statistical properties
features as revealed in the simulations. For dimensions
of the former with new tools developed in the stochastic
d = 2,...,5, there is a clear trend of narrowing of the
community [2, 5, 11].
PDFforincreasingd. Thistrendisexplainedbyexamin-
The Observable A. For a ring polymer, let r be the
i ing Eq. (1): As d increases, the noise term becomes neg-
positionoftheithmonomerinspacewherei=0..N,and
ligiblecomparedwiththeforceterm,resultinginsmaller
weplacetheoriginatthepositionofthezerothmonomer,
fluctuations and narrower tails. Against this expected
r . Fortheweaklyinteractingchainthismonomerisalso
0 trend are the results in d = 1 for the weakly interacting
the excluding one. We study a new measure, A, for the
model. Aswewillshowanalytically,theweaklyinteract-
size of a ring polymer, defined by
ing model in dimensions one and three surprisingly have
N the same distribution even though d=1 has a vanishing
(cid:88)
A= |ri−r0|. (2) deterministic force term in Eq. (1) while for d = 3 the
i=0 force is clearly not zero. In addition, we observe that for
In the RW language, A is the area under a random pro- d ≥ 3 the shape of the distribution of weakly interact-
cess, and hence is a random variable itself. Clearly ¯l = ing and ideal ring chains coincide, indicating that weak
3
interactions are negligible (when N → ∞). As we shall The denominator gives the proper normalization condi-
see,thisisalsoobservedinthetheory. Indeedthetheory tion.
discussedbelowsuggeststhatthesetwodistributionsare The first step in the calculation is to perform a simi-
already identical for d=2. However, since this is a crit- larity transformation:
ical dimension, due to extremely slow convergence, we
don’t see this behavior in the simulations. This asymp- (cid:18) r (cid:19)d−21
toticconvergenceislogarithmic(seeSM)andanεexpan- G˜t(r,s|r0)= r K(cid:101)t(r,s|r0). (5)
0
sionshowsthatitisreminiscentofcriticalslowingdown. Using Eq. (3), K˜ (r,s|r ) is the imaginary time propa-
t 0
As for the SAW polymer, we see that the fluctuations gator of a Schr¨odingier operator:
are considerably reduced compared to the other models.
TofhaisSisAWdueptoolytmheerfaisctstmhaaltletrhtehnaunmfborerthoef cootnhfiegrumraotdioenlss, HˆK(cid:101)t(r,s|r0)+ ∂∂tK˜t(r,s|r0)=δ(r−r0)δ(t) (6)
hence fluctuations are smaller. A striking observation is
with the effective Hamiltonian:
thatthetwoandthreedimensionalSAWresultsareiden-
tical, both being equal to the simulations of the d = 5 1 ∂2 (d−2)2−1
Hˆ =− + +sr. (7)
models. We now address these observations with theory. 2∂r2 8r2
Functionals of Constrained Bessel Processes. Ourgoal
The effective Hamiltonian reveals a subtle symmetry,
is to find the PDF P(A ,t) of the functional A =
B B
(cid:82)tr(t)dt of the Bessel process, constrained to start and namely two systems in dimensions d1 and d2 satisfying
0 d +d =4 behave identically. Note that this symmetry
end at the origin. We show that the difference be- 1 2
isnotaffectedbythechoiceoffunctional(orobservable)
tween the weakly interacting model (the Bessel excur-
since the latter only modifies the last term in Hˆ. This
sion) and the ideal polymer (the reflected Bessel bridge)
explains the identity of the d=1 and d=3 PDFs noted
enters through the boundary condition in the Feynman-
earlier.
Kac type of equations describing these functionals. The
Boundary Conditions for Ideal and Weakly Interacting
choice of boundary condition turns out to be non-trivial
Models. The solution of Eq. (6)
and controls the solution. Other aspects of the solution
follow the steps in [5]. (cid:88)
It is useful to find first the Laplace transform of K(cid:101)t(r,s|r0)= φk(r)φk(r0)e−λkt (8)
P(A ,t), i.e., P˜(s,t) = (cid:82)∞P(A ,t)exp(−A s)dA to k
B 0 B B B
solve the equations, and invert P˜(s,t) back to P(AB,t). is constructed [5] from the eigenfunctions φ of Hˆ where
k
Let G (r,A |r ) be the joint PDF of the random pair
t B 0 λ is the kth eigenvalue and the normalization condition
k
(r,AB)withinitialconditionG0(r,AB|r0)=δ(AB)δ(r− is (cid:82)∞φ2(r)dr = 1. The subtle point in the analysis is
r ) and G˜ = G˜ (r,s|r ) its Laplace pair. The modified 0 k
0 t 0 the assignment of the appropriate boundary condition
Feynman-Kac equation reads [18]:
correspondingtotheunderlyingpolymermodelswecon-
sider. The eigenfunctions at small r exhibit one of two
1(cid:20) ∂2 ∂ 1−d(cid:21) ∂
+ G˜−srG˜ = G˜. (3) behaviors:
2 ∂r2 ∂r r ∂t
φ+ ∼d+r1+|22−d| or φ− ∼d−r1−|22−d|. (9)
withG˜| =δ(r−r )andr acutoffwhichiseventually k k k k
t=0 0 0
taken to zero. For d = 1, the second term on the right From the normalization condition, the φ− solution can-
hand side vanishes, and we get the celebrated Feynman- not be valid for d≥4 and d≤0. For the critical dimen-
KThaceetqhuiradtiloinnecaorrtreersmpo−ndsirnG˜gtsoteBmrsowfrnoimantfhuencchtiooincealosf[1o7u]r. sdi−konr12dln=r.2Wtheentwowo ssoollvuetiothnesparroe:blφem+k ∼ford+kthre12twoorboφu−knd∼-
observable, namely our functional AB is linear in r [18]. ary conditions andthen show how to choose the relevant
Sincewearedescribingaringpolymer,theBesselprocess one for the physical models under investigation.
must start and end on the origin, and so, following [10], The distribution of A/(cid:104)A(cid:105). Following the Feynman-
we need to calculate Kac formalism described above and performing the in-
G˜ (r,s|r ) verse Laplace transform [5], we find two solutions for
P˜(s,t)= lim t 0 . (4) the PDF of the scaled variable χ≡A/(cid:104)A(cid:105)
r=r0→0G˜t(r,0|r0)
4
Γ(1±|α|)(cid:18) 4 (cid:19)±|α|+1
p±(χ)=− √
2πχ ( 2c χ)2/3
±
×(cid:88)∞ [d˜±]2(cid:20)Γ(cid:18)5 ±|ν|(cid:19)sin(cid:18)π2±3|ν|(cid:19) F (cid:18)8 ± |ν|,5 ± |ν|;1,2;− 2λ3k (cid:19)
k 3 3 2 2 6 2 6 2 3 3 27(c χ)2
±
k=0
λ (cid:18)7 (cid:19) (cid:18) 4±3|ν|(cid:19) (cid:18)7 |ν| 5 |ν| 2 4 2λ3 (cid:19)
− √ k Γ ±|ν| sin π F ± , ± ; , ;− k
( 2c±χ)2/3 3 3 2 2 6 2 3 2 3 3 27(c±χ)2
1(cid:18) λ (cid:19)2 (cid:18) |ν| 3 |ν| 4 5 2λ3 (cid:19)(cid:21)
+ √ k Γ(3±|ν|)sin(±π|ν|) F 2± , ± ; , ;− k . (10)
2 ( 2c±χ)2/3 2 2 2 2 2 3 3 27(c±χ)2
The solution is independent of N and valid in the limit d φ+ J+ φ− J−
k k k k
of N → ∞. Here, |α| = |d − 2|/2, |ν| = 2|α|/3, and
d=1 φ+ ∼r J+ <0 φ− ∼Const J− =0
F (·)referstothegeneralizedhypergeometricfunctions. k √ k k √ k
2 2 d=2 φ+ ∼ r J+ =0 φ− ∼ rlogr J− >0
The supplementary material provides a list of λ and d k k k k
k k d=3 φ+ ∼r J+ =0 φ− ∼Const J− >0
values for d=1,...4. For d=1 the solution agrees with k k k k
theknownresults[10,11,20,21], wherethe+solutionis
TABLE I. Behavior of the probability eigenfunctions and
the celebrated Airy distribution [10, 11]. The average of probabilitycurrentsintheproximityoftheorigin(r=0+)for
A is the +/− solutions in different dimensions. A current J− >0
ontheoriginisunphysicalhencethecriticaldimensionwhere
(cid:16) (cid:17)
πΓ ±|2−d| + 3 local interactions are uniportant is 2.
(cid:104)A(cid:105)± =c±N3/2, c± = √ (cid:16) 2 2 (cid:17). (11)
4 2Γ ±|2−d| +1
2
theidealandweaklyinteractingpolymermodelswithour
The + solution was previously presented in a slightly theoretical results for P+(A/(cid:104)A(cid:105)), as given in Eq. (10).
different form in [5] and here the question is how to As noted above, for d ≥ 3, we see that even for finite
choose the solution for the corresponding polymer mod- size chains the local interaction is not important, and
els. Clearly, for d = 2, (cid:104)A(cid:105)+ = (cid:104)A(cid:105)−, indicating that that the theory and simulations perfectly match, while
this is a critical dimension. Further (cid:104)A(cid:105)+ in 1 and 3 for d=2 there are strong finite size effects in the weakly
dimensions are identical and so is (cid:104)A(cid:105)− as the result of interacting case.
the symmetry around d = 2 in Eq. (7). The scaling Self-avoiding polymers. Extensive simulations of ring
(cid:104)A(cid:105) ∝ N3/2 is expected since r scales with the square SAWs were performed on cubic lattices. As has already
root of N as for Brownian motion, so the integral over been pointed out, the global expansion of a polymer is
the random processes r scales like N3/2. characterized by the exponent ν . Since A constitutes
p
We investigate the physical interpretation of the a measure of the overall size of a polymer, its behavior
two possible boundary conditions. A mathematical for large N should follow A∼Nνp+1. For the ideal and
classification of boundary conditions was provided in weakly-interacting chains ν = 1/2, and A ∼ N3/2. For
p
[15, 19] and here we find the physical situations where SAWstheexactvalueoftheexponentdependsond,i.e.,
these conditions apply. We examine the behavior of the ν = ν (d). ν = 1,0.75, and 0.5 are known exactly
p p p
probability current associated with the kth mode: J± = for d = 1,2, and 4 [22], respectively, while for d = 3,
−1φ±(r )(cid:0)(r(d−1)/2φ±(r))(cid:48)+(1−d)r(d−1)/2φ±(r)/rk(cid:1) ν , based on renormalization group considerations and
2 k 0 k k p
for r near the boundary r → 0 in dimension d. The MonteCarlosimulations,isν (cid:39)0.588. Theseprediction
0 p
analysis is summarized in Table I. We see that in were extensively tested numerically for the observable of
dimension two and higher, the current on the origin interest A with a critical dimension of d = 4, character-
is either zero or positive. A positive current at the izedbyaveryslowconvergenceoftheweakly-interacting
boundary means that probability is flowing into the model (see SM).
system, which is an unphysical situation in our system. WhilethescalingbehavioroftheSAWmodelisdiffer-
Henceweconcludethatindimensiontwoandhigher,the entfromthatoftheothertwomodels(asreflectedinν ),
p
− solution is not relevant. This implies that statistics of as we have noted, the scaled PDFs, P(A/(cid:104)A(cid:105)) are never-
excursion and reflected bridges (and equivalently, ideal theless similar. A striking observation is that the SAWs
and weakly interacting ring polymers) are identical for in d = 2 and 3 coincide to the precision of our measure-
d≥2 and correspond to the + solution. ments with the comparably narrow PDF of the d = 5
In Fig. 1 we compare the results of the simulations of non-interactingmodel, (seeFig. 1). Thatthesedistribu-
5
tion are narrower than the non-self-avoiding case can be (15) E.Martin,U.Behn,andG.Germano,Phys. Rev. E83,
qualitatively explained as follows: Since the interaction 051115 (2011).
forbids many compact conformations, the fluctuations of (16) E. Barkai, E. Aghion, and D. A. Kessler, Phys. Rev. X
4, 021036 (2014).
the area become smaller. This is easily observed in the
(17) S. N. Majumdar, Current Science 89, 2076 (2005).
extreme case of a linear SAW in one dimension where
(18) S.Carmi,andE.Barkai,Phys.Rev.E84,061104(2011),
only one conformation is allowed and the scaled PDF
(19) J. Pitman, and M. Yor Probability Theory and Related
assumes the form of a δ-function. Fields 59, 425 (1982).
Discussion. The mapping of ring polymer models to (20) L. A. Shepp, Ann. Prob. 10, 234 (1982).
(21) F. B. Knight, Intl. J. of Stoch. Analysis 13, 99 (2000).
the reflected Bessel bridge and excursion is very promis-
(22) B. Nienhuis, Phys. Rev. Lett. 49, 1062, (1982).
ingsinceitimpliesthatnotonlytheobservableAcanbe
analyticallycomputed, butalsoothermeasuresofstatis-
tics of ring polymers. An example would be the max-
imal distance from one of the monomers to any other
monomer,sincethatwouldrelatetoextremevaluestatis-
tics of a correlated process. The famed Airy distribution
describes both the one dimensional polymer, as well as
the three dimensional one, due to the symmetry we have
foundintheunderlyingHamiltonian. Thecaseofd=2is
criticalinthesensethatinteractionontheoriginbecomes
negligible, though for finite size chains it is still impor-
tant. BoundaryconditionsoftheFeynman-Kacequation
were related to physical models, which allowed as to se-
lectthesolutionsrelevantforphysicalmodels. TheSAW
polymer exhibits interesting behavior; the distribution
of A/(cid:104)A(cid:105) is identical (up to numerical precision) in di-
mension 2 and 3 and corresponds to the non-interacting
modelsindimension5. Furtherworkonthisobservation
is required.
Acknowledgments: ThisworkissupportedbytheIsrael
Science Foundation (ISF).
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S6
1
0.8
2
3/
N
/0.6
æA
Æ
0.4
0.2
1 10 100 1000 10000 1e+05 1e+06 1e+07
N
FIG. S1. (color online) (cid:104)A(cid:105)/N3/2 as a function of N for the critical dimension d=2: weakly-interacting polymer simulations
(blue circles) and the fitted curve of the function: (cid:104)A(cid:105)/N3/2 (cid:39)0.497+0.544/log(N). The black dashed line is the theoretical
value of c+ =0.4922 for N →∞.
SUPPLEMENTARY MATERIAL
Ring Polymer Simulations in d dimensions
Ideal and Weakly Interacting Polymers
In our simulations for the ideal and weakly-interacting ring polymers the chain is built of N consecutive bonds
on a lattice. In the ith step, the bond displacement is ∆r = ±1 in each of the d directions, j = 1,...,d, hence
i,j
(cid:113) √
the bond length is b = (cid:80)d (∆r )2 = d. For example, for d = 3, starting from the origin (i = 0) with first
j=1 j √
step of ∆r = (+1,−1,+1) (yielding b = 3) we reach the lattice site r = (+1,−1,+1). For a second step of
1
∆r=(−1,−1,−1) we end up at lattice site r =(0,−2,0) for the i=2 monomer.
2
In order to maintain the closure condition of the chain, we choose an array of length of N with N/2 components of
(+1)andN/2of(−1)ineachofthedirections(dsucharrays),andthenshufflethemforeachdirectionseparately. For
eachi,thecomponentsofourddimensionalsteparetheithvaluesofthesearrays. Thesumofallofthedisplacements
in each direction is then naturally zero so that the last monomer is always positioned at the origin. For the ideal
chain model we built 106 such conformations while for the weakly interacting we threw away all the conformations
that crossed the origin prior to the final monomer. For each of the conformations we calculated A = (cid:80)N |r −r |,
i=0 i 0
where r =0, and plotted the distribution of this parameter.
0
According to Eq. (11) in the paper, for N → ∞ we have (cid:104)A(cid:105) = c N3/2. By this we can check the convergence
±
of the simulations to the theory as a function of N. At the critical dimension, d = 2 this convergence becomes very
slow. In Fig. S1 we plot (cid:104)A(cid:105)/N3/2 for different N values (in a logarithmic scale) in d=2, where (cid:104)A(cid:105)=c =0.4922
+
is the theoretical value for N →∞. One can observe the very slow logarithmic convergence to this value.
Self-Avoiding Ring Polymers
Monte Carlo (MC) simulations have been applied to self-avoiding ring polymer models [1] on square, simple cubic,
and d = 4 hyper-cubic lattices. The polymer consists of N monomers (and N bonds), where the first monomer
is attached to the origin of the coordinate system on the lattice, and the Nth monomer is the nearest-neighbor to
the origin. At step j of the MC process, monomer k (1 ≤ k ≤ N −1) is selected at random (i.e., with probability
1/(N −1)) and the segment of m monomers following k (i.e., k+1,k+2,,k+m) become subject to change in the
S7
FIG.S2. (coloronline)TheoreticalP(A/(cid:104)A(cid:105))forreflectedBesselbridgesandBesselexcursionsford=4(blacksolidline)along
with simulations of d = 4 SAW for N = 200 (blue diamonds), N = 400 (green triangles), N = 800 (blue circles), N = 1600
(purple squares).
MC process; the rest of the chain (i.e., monomers 1 to k and k+m+1 to N) is held fixed (notice that if k is at the
end of the chain, N−m+1<k ≤N−1, m decreases correspondingly from m−1 to 1). Thus, this current segment
is temporarily removed and a scanning procedure is used to calculate all the possible segment configurations of m
monomerssatisfyingtheexcludedvolumeinteractionandtheloopclosurecondition(i.e.,thesegmentofmmonomers
should start at k and its last monomer, k+m is a nearest neighbor to monomer k+m+1; notice that the initial
segment configuration is generated as well). The segment configuration for step j is chosen at random out of the set
of L configurations generated by the scanning procedure and the MC process continues.
This process starts from a given ring configuration, whose transient influence is eliminated by a long initial simula-
tion, which leads to typical equilibrium chain configurations. Then, every certain constant amount of MC steps the
currentringconfigurationisstoredinafiletocreateafinalsampleofnringsfromwhichtheaveragesandfluctuations
of the physical properties of interest are calculated. The segment sizes used are m=10 for the square lattice, m=6
for the simple cubic lattice, and m=4 for d=4. For each lattice several chain lengths, N are studied.
We calculated the averages of R2,
g
N−1
1 (cid:88)
R2 = (r −r )2, (S1)
g N i c.m.
i=0
where
N−1
1 (cid:88)
r ≡ r (S2)
c.m. N i
i=0
(cid:113)
and A = (cid:80)Ni=−01|ri−r0|. For large N, (cid:104)Rg(cid:105) = (cid:10)Rg2(cid:11) increases as ∼ Nνp where νp is a critical exponent and as
discussed in the text, ν = ν +1. However, we are mainly interested in the fluctuations of A, i.e., in the shape of
A p
thescaleddistributionofA/(cid:104)A(cid:105)fordifferentN. TocheckthereliabilityofthesimulationsweprovideinTableS1the
resultsobtainedforR . Theν resultsford=2andd=3areequalwithintheerrorbarstothoseofν
g calculated predicted
while for d = 4 ν is too large due to a logarithmic correction to scaling, which would become insignificant
calculated
only for much larger N. In fact, considering this correction in the analysis has led indeed to ν (cid:39) 0.5. The
calculated
same quality of results for ν has been obtained for the observable A (where ν (cid:39)1.5 for d=4).
A A
The d=4 SAW case is a critical one, since the critical exponent, ν for lower dimensions significantly differs from
p
the ν =1/2 of the non-interacting models, and for d≥4 the interactions become unimportant for N →∞. Hence,
p
we expect P(A/(cid:104)A(cid:105)) of the d=4 SAW to coincide with that of the non-interacting model. However, for finite N the
S8
d N range ν ν
calculated predicted
2 200−1600 0.7507±0.002 0.75 (exact)
3 60−1002 0.5890±0.002 0.588
4 200−2560 0.525±0.015 0.5 (exact)
TABLE S1.
Dimension Value k=1 k=2 k=3 k=4 k=5 k=6 k=7
1,3 λ+ 2.3381 4.088 5.2056 6.7871 7.944 9.02265 10.040
k
d+ 1 1 1 1 1 1 1
k
1 λ− 1.0188 3.2482 4.8201 6.1633 7.3722 8.4885 9.5345
k
d− 0.99088 0.5550 0.4554 0.4241 0.3837 0.3663 0.3566
k
2 λ+ 1.738 3.671 5.170 6.475 7.658 8.755 9.787
k
d+ 1.1391 0.9195 0.8386 0.7885 0.7597 0.7317 0.7109
k
4 λ+ 2.873 4.494 5.868 7.098 8.231 9.291 10.294
k
d+ 0.7585 0.8807 0.9523 1.0047 1.0482 1.0820 1.1108
k
5 λ+ 3.362 4.885 6.208 7.406 8.516 9.558 10.547
k
d+ 0.5187 0.6762 0.7838 0.8646 0.9405 1.0012 1.0531
k
TABLES2. Thefirst7eigenvalues,λ andthecorrespondingnumericcoefficientsd requiredforplottingthetheoreticalPDFs.
k k
The eigenvalues λ+ of the + solutions for d=1 and d=3 are the negatives of the zeros of the Airy function: Ai(−λ+)=0.
k k
The eigenvalues λ− of the − solution are the negatives of the zeros of its derivative: Ai(cid:48)(−λ−) = 0. The d− for d = 1 are
k k k
tabulated in Ref. [3].
interaction still has an effect on the distribution’s shape, and an even more pronounced one for ring polymers. For
the values of N we used in our SAW simulations the curve had not yet converged as can be seen in Fig. (S2). A
downward trend of the curves towards that of the non-interacting case (i.e. towards convergence) can nevertheless be
seen. A similar problem is not found for SAW in d=2,3 which are reported in the main text.
Numeric values of λ and d
k k
The theoretical PDFs for different dimensions, presented in Eq. (10) in the paper, may be plotted using
MATHEMATICA(cid:114). In order to find the numerical coefficients λ (eigenvalue) and d (the normalization coeffi-
k k
cient of the eigenfunction) values of the kth mode, we used the numerical method described in detail in [2]. In Table
(S2) we present the values of the first few λ and d for different boundaries in different dimensions. We found that
k k
the first 7 eigenvalues are usually sufficient for the evaluation of P±(A/(cid:104)A(cid:105)).
(1) H. Meirovitch, J. Chem. Phys. 89, 2514 (1988).
(2) E. Barkai, E. Aghion, and D. Kessler, Physical Review X 4, 021036 (2014).
(3) M.AbramowitzandI.A.Stegun,Handbook of mathematical functions: with formulas, graphs, and mathematical tables,55
(Courier Dover Publications, 1972).