April 8, 2004

Billiards, random matrices, M-theory and all that

Posted by Urs Schreiber

I am currently at a seminar on quantum chaos and related stuff. You cannot enjoy meetings like these without knowing and appreciating the Gutzwiller trace formula which tells you how to calculate semiclassical approximations to properties of the spectrum of chaotic quantum systems (like Billiards and particles on spaces of constant negative curvature) by summing over periodic classical paths.

One big puzzle was, and still is to a large extent, why random matrix theory reproduces the predictions obtained by using the Gutzwiller trace formula.

In random matrix theory you pick a Gaussian-like ensemble of matrices (orthogonal, symplectic or unitary ones) and regard each single such matrix as the Hamiltonian operator of some system. It is sort of obvious why this is what one needs for systems which are subject to certain kinds disorder. But apparently nobody has yet understood from a conceptual point of view why it works for single particle systems which are calculated using Gutzwiller’s formula. But there is quite some excitement here that one is at least getting very close to the proof that Gutzwiller does in fact agree with RMT, see

Stefan Heusler, Sebastian Müller, Petr Braun, Fritz Haake, Universal spectral form factor for chaotic dynamics (2004) .

One hasn’t yet understood why this agrees, only that it does so. My hunch is that it has to do with the fact that by a little coarse graining we can describe the classical chaotic paths as random jumps and that the random matrix Hamiltonians are just the amplitude matrices which describe these jumps.

But anyway. ‘Why all this at a string coffe table?’, you might ask.

Well, while hearing the talks I couldn’t help but notice the fact that I actually do know one apparently unrelated but very interesting example of a system which, too, is described both by chaotic billiards as well as by random matrices. This system is - 11 dimensional supergravity.

I had mentioned before the remarkable paper

T. Damour, M. Henneaux , H. Nicolai Cosmological Billiards (2002)

where it is discussed and reviewed how theories of gravity (and in particular of supergravity) close to a spacelike cosmological singularity decouple in the sense that nearby spacetime points become causally disconnected and how that leads to a mini-superspace like dynamics in the presence of effective ‘potential’ walls’ which is essentially nothing but a (chaotic) billiard on a hyperbolic space.

(This paper is actually a nice thing to read while attending a conference where everybody talks about billiards, chaos, coset spaces, symmetric spaces, Weyl chambers and that kind of stuff. )

So 11d supergravity in the limit where interactions become negligible is described by a chaotic billiard just like those people in quantum chaos are very fond of.

But here is the crux: 11d supergravity is also known to be approximated by the BFSS matrix model. Just for reference, this is a system with an ordinary quantum mechanical Hamiltonian

(1)$H = \mathrm{Tr}\left( \frac{1}{2}\dot X^i \dot X_i - \frac{1}{4}\left[X^i,X^j\right] \left[X_i,X_j\right] \right) + {fermionic terms} \,,$

where the $X^i$ are large $N \times N$ matrices that describe D0-branes and their interconnection by strings or, from another point of view, blobs of supermembrane.

Hm, but now let’s again forget about the interaction terms. Then the canonical ensemble of this system is formally that used in random matrix theory!

Am I hallucinating ot does this look suggestive?

I think what I am getting at is the following: Take Damour&Henneaux&Nicolai’s billiard which describes 11d supergravity. Now look at its semiclassic behaviour. It is known that this is governed by random matrix theory (But we have to account for some details, like the fact that the mini-superspace billiard is relativistic. Maybe we have to go to its nonrelativistic limit.) We realize that the weight of the random matrix ensemble is the free kinetic term of the BFSS model. Therefore we might be tempted to speculate that the true ensemble of randowm matrices which is associated with 11d supergravity away from the cosmological singularity is obtained by including the $[X,X]^2$ interaction term of the BFSS Hamiltonian in the weight. With this RMT description in hand, try to find the corresponding billiard motion. Will it coincide with the speculation made by DHN about the higher-order corrections to their mini-superspace dynamics?

In any case, I see that apparently random matrix theory (‘like every good idea in physics’ ;-) has its place in string theory. I should try to learn more about it.

Posted at April 8, 2004 12:34 AM UTC

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Random matrices, M-theory and black holes

Several years ago I looked at the role of random matrices in Matrix theory black holes. Eventually, I wrote a paper, Eigenvalue Repulsion and Matrix Black Holes, in which I argued that the eigenvalue repulsion displayed by random matrices gives rise to the holographic behavior of matrix black holes. I did not show this, but I did argue it.

The paper would probably be a pretty light read for you, but it might fit with your interest in the connection between Matrix theory and random matrices. It does not address any cosmological issues.

Cheers,
Gavin

Posted by: Gavin Polhemus on April 9, 2004 8:48 PM | Permalink | Reply to this

Re: Random matrices, M-theory and black holes

Hi -

thanks for the nice link! Very interesting.

Let me see if I correctly understand what you are saying in this paper:

- - You point out that as D0-branes approach each other (to within the ‘stadium’) the non-diagonal matrix elements become comparable to the diagonal ones.

- - This means that here we can roughly regard the corresponding matrix sub-blocks as being picked from an RMT ensemble.

- - This means that the eigenvalues will be equally spaced, on average, with the spacing being of order of ${l}_{P}$ (Planck’s length).

- - Since the eigenvalues are nothing but the effective D0 brane coordinates this means that the D0 branes will on avarage occupy a volume of side-length ${l}_{P}$, at least. This effect is not due to a potential but due to the statistical RMT effect.

So if I understand this correctly this gives a nice connection between statisics of RMT and the lower bound ${l}_{P}$ on any physically meaningful distance.

Posted by: Urs Schreiber on April 11, 2004 6:06 PM | Permalink | PGP Sig | Reply to this

Re: Random matrices, M-theory and black holes

Your summary is exactly right. I’d word the final conclusion a bit differently. Eigenvalue repulsion gives a physical explanation for the density bound of one D0-brane per Planck volume. It also suggests an explanation for why Boltzman statistics should be used to calculate back hole thermodynamics. The presence of off diagonal elements between neighboring D0-branes breaks the permutation symmetry, so the D0-branes become distinguishable.

When I was working on my Ph.D. I wrote this paper and another which used RMT to explain why coincident wrapped D-branes join up to become a single D-brane with multiple winding. In that case I was able to actually do the calculations because everything was happening in just one dimension. The resulting D-branes could carry fractional winding momentum states which were necessary to explain some black hole entropy calculations done by others at that time. (I also wrote a matrix theory potential paper with Robert Helling during that time, but it had nothing to do with RMT.)

The eigenvalue repulsion idea was picked up by a few people, including Leonard Susskind in a paper on Matrix Theory Black Holes and the Gross Witten Transition. The idea was nice, but I didn’t feel that we were ever able to back it up with a real calculation when the D-branes were able to move in more than one dimension.

I got out of active research after those papers, but I’m returning to it now. I’m quite glad to have this coffee table and the new sci.physics.strings news group to help me get back into the conversation.

Cheers,
Gavin

Posted by: Gavin Polhemus on April 12, 2004 4:49 AM | Permalink | Reply to this

I messed up the first link in my post. It should go to Statistical Mechanics of Multiply Wound D-Branes.

Posted by: Gavin Polhemus on April 12, 2004 4:55 AM | Permalink | Reply to this

Re: Random matrices, M-theory and black holes

I wrote:

lower bound on any physically meaningful distance

You wrote:

I’d word the final conclusion a bit differently. Eigenvalue repulsion gives a physical explanation for the density bound of one D0-brane per Planck volume.

Ok, I see. But I was thinking of the fact that as the originally diagonally-dominated matrices turn into members of a GUE with lots of off-diagonal contributions as the D0-branes approach each other. So the eigenvalues stay at distances of at least ${l}_{P}$ (in the mean), but the notion of what the D0-brane is at that eigenvalue coordinate changes since the matrices are now diagonal in a different basis. So it seems that as one tries to push two D0-branes closer than ${l}_{P}$ they kind of dissolve and ‘different’ D0 branes pop up a distance ${l}_{P}$ apart. This is of course a little different from, say, a hard-core repulsion, which would also give just the density bound. That’s why I phrased it the way I did.

You kindly pointed me to the interesting paper hep-th/9612130.

What I don’t quite see yet is how the probability measure on elements of $\mathrm{SU}\left(N\right)$ in equation (5) is related to the Gaussian measure on the Lie algebra elements. Are they equivalent?

Posted by: Urs Schreiber on April 14, 2004 6:45 PM | Permalink | PGP Sig | Reply to this

Matrix Measures

Since SU(N) is a compact group, it is natural to use the measure that is invariant under the action of the group. I don’t worry about the Lie algebra because the connection (valued in the Lie algebra) doesn’t play any role in the calculation. It is only the Wilson line (valued in the group) that has physical significance.

I don’t think this measure is the same as the Gaussian measure on the Lie algebra. The Lie algebra is non-compact, so if you want to have a finite measure you need to have it trail off as you get far from the origin. If I remember correctly, the Gaussian measure does this with an exp(-Tr(X^2)) coefficient. This measure is not invariant under the group. It is appropriate when the group has been broken by Wilson line in another dimension. These Wilson lines generate a X^2 potential, so you wouldn’t expect the measure to be the same.

I hope this addressed your question. Do I understand the Gaussian measure correctly?

Gavin

Posted by: Gavin Polhemus on April 16, 2004 3:06 PM | Permalink | Reply to this

Interpretation of RMT ensemble for single systems

Stefan Heusler and Sebastian Müller told me (if I understood them correctly) that the importance of their result nlin.CD/0309022 is that it makes progress towards elucidating why Random Matrix Theory, which seems to be about ensembles of physical systems, makes such good predictions about the spectral statistics of single chaotic systems.

Indeed, in the introduction of the above paper it says:

Understanding the observed fidelity of individual dynamics to RMT has been an elusive goal, in spite of considerable efforts […]. We shall here take a non-trivial step towards that goal, on semiclassical ground.

As I said above, I am interested in understanding this relationship because I feel that it may help see a conceptual connection between the BFSS Matrix Model description of supergravity with the one in terms of chaotic cosmological billiards by Henneaux,Damour&Nicolai.

To me it seems that it should be possible to understand why (conceptually) RMT has something to say about, for instance, chaotic billiards, over and above checking that the trace formula over periodic orbits does indeed reproduce the RMT prediction for the spectral ‘form factor’.

While visiting my and my gilrfriend’s parents over Easter holidays and while sipping coffee and eating easter eggs ;-) I had the leisure to spend a couple of thoughts on this question. These are what I would like to discuss in the following.

The key idea is the following: Stefan Heusler had emphasized to me that the problem is that in a single system it is not clear what should correspond to the ensemble used in RMT. I want to discuss the possibility that if we consider a coarse-graining of the phase space of the system into cells large enough that all non-universal behaviour takes place only within a single such cell, while the coarse-grained large-scale behaviour is truly chaotic, that then the ensemble of the RMT corresponds simply to the set of phase space points within a single such cell.

I would like to demonstrate some circumstantial formal evidence for this claim, which, while being short of a being proof, even in the physicist’s sense, should make the conjecture quite plausible.

So pick some classically chaotic system $S$ with classical phase space ${P}_{S}$ and partition phase space into disjoint cells ${C}_{i}$ with $i\in \left\{1,2,3,\cdots ,N\right\}$ such that

(1)${P}_{S}=\bigcup _{i}{C}_{i}\phantom{\rule{thinmathspace}{0ex}}.$

Consider furthermore the set of all $N×N$ matrices of pairs of phase space points to any given pair of such cells, e.g.

(2)$\lambda =\left[\begin{array}{cccc}\left({x}_{11},{y}_{11}\right)& \left({x}_{12},{y}_{12}\right)& \cdots & \left({x}_{1N},{y}_{1N}\right)\\ \left({x}_{21},{y}_{21}\right)& \left({x}_{22},{y}_{22}\right)& \cdots & \left({x}_{2N},{y}_{2N}\right)\\ ⋮\\ \left({x}_{N1},{y}_{N1}\right)& \left({x}_{N2},{y}_{N2}\right)& \cdots & \left({x}_{\mathrm{NN}},{y}_{\mathrm{NN}}\right)\end{array}\right]$

where ${x}_{\mathrm{ij}}\in {C}_{i}$ and ${y}_{\mathrm{ij}}\in {C}_{j}$.

These matrices give rise to approximations to the exact Hamiltonian $H$ of our system by the following procedure:

For a given point $x\in {P}_{S}$ let $|x⟩$ be a state in the Hilbert space of the system which is of minimal uncertainty (a generalized coherent state) and peaked at values of coordinates and momenta given by $x$. (Note that $x$ denotes a point in phase space.)

Now construct all $N×N$ matrices ${h}_{\lambda }$ indexed by the above matrices $\lambda$ and defined by

(3)$\left({h}_{\lambda }{\right)}_{\mathrm{ij}}:=⟨{x}_{\mathrm{ij}}|H|{y}_{\mathrm{ij}}⟩\phantom{\rule{thinmathspace}{0ex}}.$

These matrices are essentially the transition amplitudes from a coherent state peaked at ${y}_{\mathrm{ij}}\in {C}_{j}$ to a state peaked at ${x}_{\mathrm{ij}}\in {C}_{i}$.

The idea behind all these definitions is that I want to decompose the trace over the system’s Hilbert space into a sum over the cells ${C}_{i}$ and a sum over coherent states peaked within each ${C}_{i}$. The point is that the ergodicity of the system allows to say something about averaged transition amplitudes between different cells. This information then can be used to say something about the distribution of the transition amplitdues between single points in these cells and hence about the ensemble of the ${h}_{\lambda }$

More precisely, the averaged amplitude for a system to move from somewhere within cell ${C}_{j}$ to somewhere within ${C}_{i}$ is

(4)$\sum _{{x}_{i}\in {C}_{i},{y}_{j}\in {C}_{j}}⟨{x}_{i}|\mathrm{exp}\left(\frac{tH}{hi}\right)|{y}_{j}⟩\phantom{\rule{thinmathspace}{0ex}},$

where the sum is a symbolic shorthand for a suitably normalized double integral over all coherent states peaked at ${x}_{i}\in {C}_{i}$ and ${y}_{j}\in {C}_{j}$. (These states will of course overlap into neighbouring cells.)

At this point I use the first essential piece of assumption/classical information: By the very construction of the ${C}_{i}$ they are supposed to be larger than the scales traversed by the system in a time span so short that non-universal behaviour still plays a role. This means that on very short time spans there is no transition on average between two cells ${C}_{i}$ and ${C}_{j}$. Therefore the above expression should vanish to first order in $t$, i.e.

(5)$\sum _{{x}_{i}\in {C}_{i},{y}_{j}\in {C}_{j}}⟨{x}_{i}|H|{y}_{j}⟩\approx 0\phantom{\rule{thinmathspace}{0ex}}.$

I like to think of this as saying that there is an ensemble of systems ${C}_{j}$ that evolves into an ensemble of systems ${C}_{i}$ such that the averaged transition amplitude cancels out.

In order to make this more precise note that the above condition can be reformulated in terms of the matrices ${h}_{\lambda }$ as

(6)$\sum _{\lambda }\left({h}_{\lambda }{\right)}_{\mathrm{ij}}\approx 0\phantom{\rule{thinmathspace}{0ex}}.$

Here $\lambda$ runs over all $\lambda$-matrices, as defined above.

It seems correct to think of the ${h}_{\lambda }$ of an ensemble of effective Hamiltonians each describing the transition of a particular system with a denumerable phase space to evolve from ${C}_{i}$ to ${C}_{j}$. The different ‘systems’ are of course nothing but sub-trajectories of the single system under consideration. They appear as different systems due to the coarse-graining.

With this interpretation, the above condition says that the average of each off-diagonal matrix entry over the ensemble of all ${h}_{\lambda }$ vanishes.

This is nice, because it agrees with the assumption made in RMT. So let’s see if we can similarly say something about the variance of the $\left({h}_{\lambda }{\right)}_{\mathrm{ik}}$:

To that end, consider the cell-averaged trace over the propagator:

(7)$\sum _{{x}_{i}\in {C}_{i}}⟨{x}_{i}|U\left(t\right)|{x}_{i}⟩:=\sum _{{x}_{i}\in {C}_{i}}⟨{x}_{i}|\mathrm{exp}\left(\frac{\mathrm{tH}}{ih}\right)|{x}_{i}⟩\phantom{\rule{thinmathspace}{0ex}}.$

By the above considerations the term of first order in $t$ vanishes. Let’s concentrate on the leading non-vanishing order ${t}^{2}$:

(8)$\sum _{{x}_{i}\in {C}_{i}}⟨{x}_{i}|{H}^{2}|{x}_{i}⟩\phantom{\rule{thinmathspace}{0ex}}.$

By inserting an (approximate) decomposition of unity and using the fact that the $|{x}_{i}⟩$ are approximately orthogonal (here is obviously the point in my discussion where all these considerations are currently still very vague and sketchy) this can be rewritten as something looking like

(9)$\propto \sum _{{x}_{i}\in {C}_{i}}\sum _{j}\sum _{{y}_{j}\in {C}_{j}}⟨{x}_{i}|H|{y}_{j}⟩⟨{y}_{j}|H|{x}_{i}⟩\propto \sum _{j}\sum _{\lambda }\left({h}_{\lambda }{\right)}_{\mathrm{ij}}\left({h}_{\lambda }{\right)}_{\mathrm{ji}}\phantom{\rule{thinmathspace}{0ex}}.$

(Please recall that sums over ${x}_{i}$ are supposed to denote suitably normalized integrals. The idea should be clear. I don’t want to clutter its discussion with overly exact notation at this point.)

In words: The probability to go along a closed orbit starting in ${C}_{i}$ and visiting given ${C}_{j}$ in the process is, to lowest non-vanishing order, the sum of the squares of the ensemble of transition amplitudes between ${C}_{i}$ and ${C}_{j}$.

Now I use the second crucial assumption/classical observation: It is known that the classical chaotic dynamics is such that on the coarse-granined level of the ${C}_{i}$ the time evolution is a complytely random hopping between the different cells. This should mean that the contribution to the above from closed paths visiting any of the ${C}_{j}$ must be the same, i.e.

(10)$\sum _{\lambda }\left({h}_{\lambda }{\right)}_{\mathrm{ij}}\left({h}_{\lambda }{\right)}_{\mathrm{ji}}\approx \sum _{\lambda }\left({h}_{\lambda }{\right)}_{\mathrm{ik}}\left({h}_{\lambda }{\right)}_{\mathrm{ki}}\phantom{\rule{thinmathspace}{0ex}}.$

Note that there is no sum over $j$ and $k$ here.

(One should probably assume a compact phase space in order to make this a little less vague than it is in any case.)

But this is nice: The above says nothing else but that the variance of the non-diagonal matrix elements of the ${h}_{\lambda }$, when avaraged over the full ensemble (all possible $\lambda$) is the same for all matrix elements.

This way one finds that the ensemble of the ${h}_{\lambda }$ is such that mean and variance of the matrix elements of the ${h}_{\lambda }$ have exactly the properties that agree with those postulated in RMT. (The mean and variance of the diagonal elements follow now from unitary/orthogonal/symplectic invariance.)

If one could argue now that due to the chaoticity of the system $S$ the ensemble of the ${h}_{\lambda }$ is such that it maximizes the information entropy with the given mean and variance of the matrix elements fixed, then one would have established that and how Random Matrix Theory describes single chaotic systems:

If the ideas sketched here are correct, they would indicate that we have to identify the random matrices that appear in RMT with the objects ${h}_{\lambda }$ defined above, which are Hamiltonians/transition amplitudes obtained from the full Hamiltonain of the system by restricting to certain microscopic trajectories such that they seem to describe an ensemble of systems on the phase space of coarse-grained cells ${C}_{i}$, where the ${C}_{i}$ are chosen in such a way that they precisely hide all small-scale non-universal properties of the system.

I should stop at this point. All comments and pointers to the literature are very welcome.

Posted by: Urs Schreiber on April 11, 2004 8:32 PM | Permalink | PGP Sig | Reply to this

Re: Interpretation of RMT ensemble for single systems

Nice ideas.

You should be interested by this recent paper Action Correlations and Random Matrix Theory by Smilansky and Verdene.

It’s sort of similar in spirit; what they say roughly is:
1) in classical hyperbolic hamiltonian systems a natural coarse-graining is that induced by its symbolic dynamics (if there’s one, not all such system can by described that way). In particular periodic orbits of any length are labelled in this way.

2) now, the RMT behaviour of the quantum spectrum induces a similar behaviour for the classical action spectrum of periodic orbits and vice-versa;

3) then they study the tractable example of the Baker Map.

Best,

thomas.

Posted by: thomas on April 15, 2004 2:41 PM | Permalink | Reply to this

Re: Interpretation of RMT ensemble for single systems

Thanks for the very interesting reference!

But after reading it I am left a little confused: The author’s assume that RMT is a valid description of the universal aspect of chaotical quantum system, they don’t derive or motivate this fact, or do they?

For instance right above equation (5) it says:

Ample numerical evidence supports the conjecture that the spectral statistics of ${U}_{N}$ is well reproduced by the predictions of rndom matrix theory

Then above equation (15) the

assumption of ‘spectral ergodicity’

is mentioned and on the top of p.5 it start out saying

Assuming that $⟨{Q}_{\mathrm{qm}}\left(n;N\right)⟩$ follows the RMT predictions […]

After having read this paper once it currently seems to me that they assume what I would like to see derived or at least motivated heuristically. Or am I confused about this point? (I have to admit that I have only read section 1 so far.)

Posted by: Urs Schreiber on April 15, 2004 5:29 PM | Permalink | PGP Sig | Reply to this

Re: Interpretation of RMT ensemble for single systems

You’re rightly confused, I first read that paper yesterday (and a little bit to quickly ;-)

The way I understand it is the following.

Up to their eq. (14) they haven’t used an RMT assumption yet, ie the relation between quantum energy density and classical action density is a general consequence of taking the semicalssical limit.

Then they say: if RMT holds, we should organize the phases in $N_g=\sqrt(N)$ subsets of size $\hat{N}=\sqrt(N)$ to indeed get a $K_{qm}(\tau ;N)$ whose $N\rightarrow\infty$ limit agrees with RMT.

So the question arises: to what organisation of the periodic orbits does this organisation of the phases correspond to. Having in mind to hopefully be able to reverse the argument as: in a hyperbolic system the POs are naturally partitionned in a way which does indeed imply RMT fluctuations of quantum spectrum, again through the generally valid semiclassical relation 14.

Then they derive more consequences on action correlations of the asumption of quantum RMT. That’s the end of section 1.

And then they develop their ideas of using symbolic dynamics with the Baker Map example. So all they do from that point on (section 2) is purely classical without using any RMT assumption.

And in section 4 they study action correlations in this example to see wether it agrees or not with what an quantum RMT asumption suggested would happen. Now they are only able to observe that POs behave in that way, i.e. things are consistent, but not proved. That’s what they say in their last section: “unfortunately, even using this tool [symbolic dynamics], we were not able to derive the classical correlation function using classical arguments only”.

I don’t know if their ideas could be pushed further to actually establish the reverse idea, but at least it’s a nice try I think.

Best,

thomas.

Posted by: thomas on April 16, 2004 10:19 AM | Permalink | Reply to this

Re: Interpretation of RMT ensemble for single systems

Thanks for you nice reply. I am currently using an awkward public internet access point at London Heathrow airport, so just a very brief reply:

It might be that Smilansky’s et al’s ideas could be turned into the proof that we are looking for. In particular I wonder if a good use of symbolic dynamics could improve the heuristic reasoning of my above sketch of a possible proof.

BTW, while thinking about that semi-proof some more it appears to me to be not all that bad when appropriate improvements are made:

1) One should explain why my h_lambda matrices have a spectrum which approximaes that of the full Hamiltonian. But that should follow from the fact that the matrix elments of the full Hamiltonian within each of the C_i cells are well correlated, by definition of these cells.

2) One should probably replace the coherent states which I used by simply position “eigenstates” in my above considerations.

3) Can symbolic dynamics be used to make the second part of my “proof” more precise?

Posted by: Urs on April 16, 2004 6:56 PM | Permalink | Reply to this

Re: Billiards, random matrices, M-theory and all that

There is a related discussion now at sci.physics.strings.

Posted by: Urs on April 18, 2004 8:15 PM | Permalink | Reply to this

Re: Billiards, random matrices, M-theory and all that

What I would expect is similar to 2 separate cells colliding - first making contact at one point then at several other points in varying succession. There would also be an uneven exchange between the two cells contents - (Think of two undulating water balloons colliding in slow motion).

Lance

Posted by: Lance Kutchins on October 5, 2008 3:34 AM | Permalink | Reply to this

Re: Billiards, random matrices, M-theory and all that

I may oversimplify by suggesting that the larger cells contents is one filled with dark energy.

Posted by: Lance on July 4, 2014 3:00 AM | Permalink | Reply to this

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