27 September (2026)

I asked Daniel to clarify the decomposition of distances into independent processes and he confirms first that in the dipolar basis, it works as it should, while in the vortex one, we got those extra factor 2 that break the interpretation, and require to defer to $D_{\ket{1,1}}^{\circ\circ}$ instead, which I didn't like because $\ket{1,1}$ is correlated. Daniel also pointed out that Eq. \eqref{eq:0EWfe} is a particular case for states that are diagonal in both bases. But states we're interested in (such as thermal states) are, so that wasn't the worst problem.

The point is, as he understood, that while the superpositions of two dipoles always gives a donut, the superposition of two donuts does not always give a dipole. It depends on their relative phases, it can even provide another donut. Whence the extra factor 2. The $\ket{1,1}$ is not so much for the independence as for ensuring that we enforce the final geometry, not allowing to wander into a variety of other things. I believe I see the argument though I'd like to see it in action. Anyway, it won't make it to the revision.

22:33 (CET).0EQdrR (AF).

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Now the physical meaning... we're getting close to the Tao Te Ching's supreme beauty being extremely simple: the pairs with both photons in the same dipole ($C'_{20}$ and $C'_{02}$, with weight $1-2\mathscr C'=4\mathscr{C}$) are distributed as independent photons on that dipole, while the pairs with one photon in each dipole ($2C'_{11}$, with weight $2\mathscr C'=1-4\mathscr{C}$) are distributed on the donut. The final distribution is their average.

In this sense, one can also write Eq. \eqref{eq:0E7B1} as \begin{equation}\label{eq:0ET8Y}D(d)=(1-\mathscr R')\,D^{\circ\circ}_\mathrm{ind}(d)+(1-2\mathscr R)\,D^\circledcirc_\mathrm{ind}(d)\,,\end{equation} which looks less nice but would read better, if it'd read as I want: the distribution of distances is that of independent dipoles weighted by the fraction of pairs both in the same dipole (so not interfering and being sampled independently, happens what happens) plus the distribution of independent donuts but it is not weighted by the fraction of pairs both in the same donut, which is $1-\mathscr{R}$, while we have it weighted by $(1-\mathscr R)-\mathscr R$, which is the non-interfering fraction minus the interfering one. So there is some subtlety that currently escapes me. This must have to do with wrong choices of complementary bases. Anyway, I have to board... Maybe I sort it out mentally (I'll probably sleep instead).

11:46 (CET).0E90C (AF).

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The program is fairly simple, work in the HG basis where the bimodal correlators become $$C'_{ij}\equiv\langle a_x^{\dagger i}a_x^ia_y^{\dagger j}a_y^j\rangle$$ in which case \begin{equation} \label{eq:0EoMp} 2\mathscr C'\equiv\frac{2C'_{11}}{C'_{20}+2C'_{11}+C'_{02}}\,, \end{equation} is the fraction of detected pairs with one photon in each dipole. Then the algebra is fairly straightforward, and we find the beautiful:

$$D(d)=D^{\circ\circ}_\mathrm{ind}(d)-\mathscr C'\,\frac d{16}(8-8d^2+d^4)\,e^{-d^2/2}$$

It is beautiful because if we compare it to Eq. \eqref{eq:0Eqwc}, we can see that the sign is opposite (expected, dipoles "bring" the correlations by cluttering the bosons, they are the real source of bunching) but also with half the weight, for what is otherwise the same correction $K\equiv \frac{d}{8}(8-8d^2+d^4)\,e^{-d^2/2}$, which is such that $\int K=0$.

It doesn't stop there. From there follows that: \begin{equation} \label{eq:0EEPg} D(d)=(1-2\mathscr C')\,D^{\circ\circ}_\mathrm{ind}(d)+2\mathscr C'\,D^\circledcirc_\mathrm{ind}(d)\,, \end{equation} which, with hindsigths, makes sense, and I'll have five hours in the plane to think about whether that is the Eq. (2) we should have in Bosoc instead.

To prettify this further, we need to link them, and this is: \begin{equation} \label{eq:0EWfe} 4\mathscr C+2\mathscr C'=1\,, \end{equation}

I'm back to wanting very much to absorb the 2 and work with $2\mathscr{C}$, although there was a reason why not. Now it escapes me and I wonder if it outweighs the sort of things that pop up in the airport cafeteria. Let me try: \begin{equation}\label{eq:0E7B1}D(d)=2\mathscr R\,D^{\circ\circ}_\mathrm{ind}(d)+\mathscr R'\,D^\circledcirc_\mathrm{ind}(d)\,,\end{equation} with $\mathscr{R}\equiv 2\mathscr{C}$ the fraction of pairs in the two vortices and $\mathscr{R}'\equiv 2\mathscr{C}'$ the fraction of pairs in the two dipoles.

11:25 (CET).0E06n (AF).

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I'm en route for the HPM (2026), where I'll discuss our big identity: $$D^\circledcirc_{\ketbra{\boldsymbol{\bar\alpha}}{\boldsymbol{\bar\alpha}}}=D_{\mathrm{ind}}^{\circ\circ}\,.$$ With the worst coffee in the world, at the airport, I'll try to express Eq. (2) of Bosoc \begin{equation}\label{eq:0Eqwc}D(d) = D^\circledcirc_\mathrm{ind}(d)+\mathscr C\,\frac{d}{8}(8-8d^2+d^4)e^{-d^2/2}\,,\end{equation} in the dipole basis, i.e., I want to find $$D(d)= D^{\circ\circ}_\mathrm{ind}(d)+\mathscr C\cdots\,.$$ This decomposition has been very inspiring for various aspects of the problem, and I trust it'll clarify some aspects of $\mathscr{C}$ in particular of its quality as a witness of non-classicality. The uncorrelated dipole has $\mathscr{C}=1/4$ but on the wrong geometry. That's actually the thread that pulls the whole pullover.

Note in passing $\mathscr C = t(1-t)$ for the $t$-distorted dipole.

10:54 (CET).0EDCI (AF).

26 September (2026)

The paper by Magaña-Loaiza et al. is a great occasion to make a first skirmish in many-mode extensions of our multiphoton spatial correlations, which so far was kept to two modes only. In their case, they find a moot autocorrelation peak only in their angular-position correlation of pseudothermal light (their Fig. 4B):

This is as moot as can be for bosons: bosons want to bunch, to clutter, to clump.

No, we say (with Eduardo), there are two! (for $\ell=\pm1$, there are all the petals of the gas burner for higher $\ell$).

This is precisely because they have a broad spectrum. In this case:

\begin{equation*} g^{(2)}(\Delta\theta)=1+|\gamma(\Delta\theta)|^2\quad\text{where}\quad \gamma(\Delta\theta)\equiv\frac{\sum_\ell n_\ell e^{i\ell\Delta\theta}}{\sum_\ell n_\ell}\,, \end{equation*}

A rapid—not fully checked—calculation of what would be our two-photon correlations for thermal, RPCS and Fock $\ket{1,1,1}$ states in three modes $\ell\in\{-1,0,1\}$ gives:

Compare with Fig. (0A6pq) for context and definitions (this is Glauber on the left, Daniel's $\tilde g^{(2)}$ on the right). The dashed are our $\ell=\pm1$ bimodal case.

Magaña-Loaiza et al. deal with the red curve on the left, but with $\pm15$ $\ell$ modes, not three, and thus their revival peak is 1+1/961 (so 1, indeed they don't see anything and probably don't suspect anything either) while with three modes it is 1+1/9, and a little bit visible.

I leave those here for possible future references, but repeat that they should be checked carefully: \begin{equation} \label{eq:0EOGN} \begin{aligned} g^{(2)}_{\rm th}&=\tfrac43+\tfrac49\cos\Delta\theta+\tfrac29\cos2\Delta\theta\,,\\ g^{(2)}_{\rm RPCS}&=1+\tfrac49\cos\Delta\theta+\tfrac29\cos2\Delta\theta\,,\\ g^{(2)}_{|1,1,1\rangle}&=\tfrac23+\tfrac49\cos\Delta\theta+\tfrac29\cos2\Delta\theta\,. \end{aligned} \end{equation} This is again for $\ell\in\{-1,0,1\}$.

20:55 (CET).0EshQ (AF).

25 September (2026)

One inconsequential error in Ref.  but which I want to record somewhere. States with $\mathcal{C}\neq0$ are not bimodal. We can say "they develop a bimodal character" instead. This becomes true only beyond $\mathscr{C}_\mathrm{crit}=0.1369$. In fact our unbalanced thermal state in the text clearly displays only one local maximum (it has $\mathscr{C}\approx 0.1045$).

22:35 (CET).0EYHL (AF).

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So no great conceptual discovery but at least that makes the agreement with the experimental data fabulous! We kill three birds with one stone: i) cleaned out the Fourier noise, ii) increased the contrast, iii) made it clean enough to match the theory.

I've asked Jacob to put error bars on that, which should make it even more viewable.

15:55 (CET).0Eysa (AF).

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This is the confirmation that projecting on the circle—which is what we've been doing all along—is the optimum version of this shaving idea: keeps all photons and optimizes contrast.

The thin rim recovers the projection formula, with its good old divergence: \begin{equation}\label{eq:0DFLo} D_{\rm ring}(d)=\frac{2}{\pi}\, \frac{1+2\mathcal{C}\bigl[2\bigl(1-\frac{d^2}{2r_0^2}\bigr)^2-1\bigr]} {\sqrt{4r_0^2-d^2}}, \qquad 0\leq d\leq 2r_0\,. \end{equation}

07:49 (CET).0DXx8 (AF).

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The basic point is that by shaving off photons, we are getting closer to the angular correlations on the circle itself, so it's like a cheap or brutal way of projecting them onto the unit circle. Shaving an annulus around the ring should thus work better. Not only it'll keep many more photons, it will also increase the contrast towards that of the exact projections. This might still be better (introducing less noise from the projection on a discretized mesh) and a particular case of more general "geographical" samplings.

07:13 (CET).0DXvJ (AF).

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The $R\to0$ limit of Jacob's razor blade brings us to an interesting and surprising result. Not only does the process works down to $R\to0$, where you give a very close shave, but it actually optimizes the process down to a universal result, once rescaled:

The camel visibility that is only 1.5% for the thermal distance $D_\mathrm{th}(d)$ becomes 21.4% in this limit. Shaved right down to the skin!

The limit reads $$D_R(d)\to\Phi(d/R)/R$$ so that, defining $x\equiv d/R$ and $\Phi\equiv\Phi_{\rm ind}+\mathcal{C}\Phi_K$ with:

$$\Phi_{\rm ind}(x)=\frac{x}{9\pi}\Bigl[24(2+3x^2)\arccos\frac x2-x\sqrt{4-x^2}\,(60+2x^2+x^4)\Bigr],$$

$$\Phi_K(x)=\frac{8x}{9\pi}\Bigl[12\arccos\frac x2-x\sqrt{4-x^2}\,(15-7x^2+x^4)\Bigr].$$

Not pretty, but universal.

It is bound within $0\le x\le 2$ because both photons are inside a disc of radius $R$, so $|\mathbf r_1-\mathbf r_2|\le 2R$, the maximum being reached only when both sit on opposite sides of the rim. Then

For $\mathcal{C}=\tfrac16$ (thermal case), the humps are at $x=0.401$ (where $\Phi(0.401)=0.588$) and $x=1.538$ (where $\Phi(1.538)=0.742$), while the dip is at $x=0.909$ (where $\Phi(0.909)=0.431$), from which we get the 21.4% contrast.

The analytical result comes from the Hankel general expression admitting closed-form solutions in this case: $$\begin{align} \psi_0&\to4\Bigl[\frac{J_1(q)}{q}-\frac{2J_2(q)}{q^2}\Bigr],\\ \psi_2&\to\frac{4J_3(q)}{q}\,. \end{align}$$ Mathematica does the rest. One can check that $\int_0^2\Phi_{\rm ind}=1$ and $\int_0^2\Phi_K=0$ exactly.

07:00 (CET).0DGGI (AF).

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Jacob got a nice idea of shaving off distant photons, as if doing some sort of evaporative cooling of distances, to cleanse but also magnifies correlations. He's been talking about this for days but since this was to get rid of extra rings of Fourier origin from the real world, I didn't pay too much attention. However when he showed me the actual results yesterday, I realized he had hit onto something nice:

This is Monte Carlo, but he did that straight on the experimental data and observed such immediate and strong improvements, also tightening the theory/experiment agreement. Now for the theory behind his idea...

The key is that angles and radii are independent—the intensity is a product $(r^2e^{-r^2}/\pi)[1+2a\cos(2\theta+\psi)]$—so a cut in $r$ leaves the angular law of the frame untouched. The pair law of the surviving photons is therefore \begin{equation} \label{eq:0DwxK} \tilde\rho^{(2)}_R(r_1,\theta_1,r_2,\theta_2) =g_R(r_1)\,g_R(r_2)\,\frac{1+2\mathcal{C}\cos2\Delta\theta}{(2\pi)^2}, \end{equation} with $g_R(r)=2r^3e^{-r^2}/P_R$ for $r\leq R$ and zero beyond, with $P_R\equiv1-(1+R^2)e^{-R^2}$ is the fraction of photons that are kept. Importantly, the contrast $\mathcal{C}=\tfrac16$ is untouched. In terms of Eq. \eqref{eq:0DW2X}, the distribution after the cutting is still of the type $$D_R(d)=D^{\odot,R}_{\rm ind}(d)+\mathcal{C}\,K_R(d),$$ but with new kernels: the contrast is the same but the shaving weights its contributions differently.

The kernels don't have particularly friendly expressions but some Hankel forms can be provided by Fourier transform in polar coordinates: $$\begin{align} D^{\odot,R}_{\rm ind}(d)&=d\int k J_0(kd)(\psi^R_0)^2dk\\ K_R(d)&=2d\int kJ_0(kd)(\psi^R_2)^2dk \end{align}$$ in terms of $$\begin{align} \psi^R_0(k)&=\frac1{P_R}\sum_{n\ge0}\frac{(-1)^n}{n!}\Bigl(\frac{k^2}{4}\Bigr)^{n}(n+1)\,\mathrm{P}(n+2,R^2),\\ \psi^R_2(k)&=\frac1{P_R}\sum_{n\ge0}\frac{(-1)^n}{n!}\Bigl(\frac{k^2}{4}\Bigr)^{n+1}\mathrm{P}(n+3,R^2) \end{align}$$ with $\mathrm{P}(a,x)$ the regularized lower incomplete gamma function. The $R\to\infty$ recovers Eq. \eqref{eq:0DW2X} which is just sanity check, but the $R\to0$ gives us a strong and surprising result, which I detail in the next 𝕫eet.

06:30 (CET).0D9rO (AF).

23 September (2026)

Carving correlations in distances: \begin{equation}\label{eq:0DW2X}D(d) = D^\circledcirc_\mathrm{ind}(d)+\mathscr C\,\frac{d}{8}(8-8d^2+d^4)e^{-d^2/2}\end{equation} $D^\circledcirc_\mathrm{ind}(d)$ are uncorrelated photons on the donut, with the blue distribution below, while the orange part is the correction, which integrates to zero, and piles up particles closer together or farther apart, taking them from the middle hump, resulting in the camel shape:

The maximum case $\mathscr{C}=1/2$ is shown.

18:45 (CET).0Dysa (AF).

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Daniel's illegal$^*$ theory on Natalia's experimental data:

18:38 (CET).0D4vs (AF).

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The physical meaning of $\mathscr{C}\equiv C_{1,1}/(C_{2,0}+2C_{1,1}+C_{0,2})$ is more clear written as $$\mathscr{C}=\langle n_1n_2\rangle / \langle N(N-1)\rangle$$

so that $2\mathscr{C}$ is the probability that the two detected photons come from the two vortices, i.e., they do not come from the same one. This is why $\mathscr{C}$ quantifies the contrast of the correlations, because same-vortex photons do not interfere. The classical reading of this is that $N(N-1)$ is the number of ordered pairs, i.e., distinguishing photons (which is first, which is second) from the whole set, while the numerator $n_1n_2$ is the number of cross-pairs, unordered. To compare same things, we make them unordered by multiplying by two. The ordering comes from detection. Indistinguishability is built in the state itself.

So $\ket{1,1}$ has $2\mathscr{C}=1$, i.e., all the photons come the two vortices: maximum contrast. For balanced thermal states, $2\mathscr{C}=1/3$: regardless of the intensity, 33% of the photons interfere. For RPCS, half of the photons interfere.

For $\ket{2,2}$, we have $N=4$ photons in total so $N(N-1)=4\times 3=12$ ordered pairs and $2n_1n_2=2\times2\times2=8$ cross pairs. Labelling the photons in $\la$ as 1 and 2, and those in $\ra$ as 3 and 4, the ordered pairs are: $$\{(1,2), (1,3), (1,4), (2,1), (2,3), (2,4), (3,1), (3,2), (3,4), (4,1), (4,2), (4,3)\}$$ while the cross pairs are: $$\{(1,3), (1,4), (2,3), (2,4), (3,1), (3,2), (4,1), (4,2)\}$$ so $2\mathscr{C}=8/12=2/3$, 66% of the photons come from different vortices. If you select one from the top list, the probability it is in the bottom list is 2/3.

The classical bound $2\mathscr{C}\leq 1/2$ says that a classical field can not deliver more than half its pairs as cross pairs. It follows from the positivity of $P$. Calling $u\equiv|\alpha_1|^2$ and $v\equiv|\alpha_2|^2$, from the normally-ordered form of $C_{ij}$, we have: \begin{equation} \label{eq:0Do7E} C_{20}=\int P(\boldsymbol\alpha)\,u^2\,d\boldsymbol\alpha\,,\quad C_{02}=\int P(\boldsymbol\alpha)\,v^2\,d\boldsymbol\alpha\,,\quad C_{11}=\int P(\boldsymbol\alpha)\,uv\,d\boldsymbol\alpha\,, \end{equation} with $d\boldsymbol\alpha\equiv d^2\alpha_1d^2\alpha_2$. Therefore, from $u^2+v^2-2uv=(u-v)^2\ge0$, the integral of this non-negative quantity over a positive $P$ remains positive, so that \begin{equation} \label{eq:0Ds7o} C_{20}+C_{02}\ge2C_{11}\,, \end{equation} i.e., $\langle{:}\hat n_1^2{:}\rangle+\langle{:}\hat n_2^2{:}\rangle\ge2\langle\hat n_1\hat n_2\rangle$: a classical field cannot deliver more (orderedd) pairs of photons with one photon in each vortex than pairs with both photons in the same vortex. The denominator of $\mathscr C$ is then at least $4C_{11}$, whence $\mathscr C\le1/4$, with equality iff $u=v$ wherever $P\neq0$, i.e., for equal intensities in every realization, as is the case for the balanced RPCS. This does not apply for states without a positive $P$.

Equation (\ref{eq:0Ds7o}) is the arithmetic-mean version of the Cauchy–Schwarz inequality nonclassicality, $C_{11}^2\le C_{20}C_{02}$. It follows from it since $2\sqrt{C_{20}C_{02}}\le C_{20}+C_{02}$, so that CSI $\implies$ Eq. \eqref{eq:0Ds7o}.

17:58 (CET).0DUn7 (AF).

22 September (2026)

Our last discussion on bosonization was 25 days ago.

I was suggesting then to consider whether the apparent bosonization problems with magnons on the one hand and Frenkel excitons on the other hand were a good starting point to further this question. Magnons don't involve bound states—composite objects—which seem to be at the heart of the problem, but Frenkel excitons do! And the mathematical shape this takes could connect Haldane and Combescot. The description by Misha was as follows:

Again, let me recall something you know quite well. In Heisenberg ferromagnet, for instance, you start with the spin Hamiltonian […](I am not discussing approximations that result in this Hamiltonian, but I can, if you really insist on revising approximations behind band structure of semiconductors/insulators) and you can find its ground state exactly (all spins are aligned along a given axis z). The problem appears when we start looking for the excited states. While for isotropic Heisenberg model the first excited states can be explicitly found (the excited state is a Bloch wave composed of states with one spin being flipped), its operators are not trivial. Their commutation relation is not exactly bosonic. Namely, the commutator contains the $S_z$-spin operator which is replaced by a number. Without such a replacement, the magnon operators are not, strictly speaking, Bose operators. Moreover, there is magnon-magnon interaction that appears directly from the Heisenberg Hamiltonian.

[…]The situation is even worse for antiferromagnets where additional problems appear as a result of having two sublattices. There are several approximations, the key one is, as I have mentioned above and also in my previous e-mail, the representation of spin operators via boson operators using Holstein-Primakoff or similar ansatz and, basically, large-spin expansion. I am ready to question and discuss these approximations and, frankly, I do not see why these approximations are better or worse than those in the theory of Wannier-Mott excitons.

Furthermore, let us look at Frenkel excitons within a trivial picture of a set of two-level atoms (or molecules) with weak tunnel coupling. Ground state of such insulator corresponds to all atoms being the in the ground state. First excited state (single exciton state) correspond to the Bloch wave of the states where one atom is in the excited state. The Hamiltonian is very similar to that of the ferromagnet in strong magnetic field. This analogy has been explored and exploited in detail by Agranovich and Toshich [1] — they explicitly mention it.

Basically, a similar issue with commutation relations appears for 1D Luttinger liquid where one recasts density operators via bosonic ones: at some point you replace complicated commutation relation by a much simpler bosonic one.

That was in his 13 August (2026) email, point 2 there of the many he made then. He comes back to the Combescot and Pogosov paper plus two others in their wake. He observes that while they present the problem as dealt by others as complicated, her conclusions are eventually the same as the "canonical" treatment:

She later writes:

The generalization [23] of this approach to an arbitrary number of excitons turns out to be rather complicated technically, in contrast to our method which is quite straightforward for any number of Frenkel excitons, thanks to equations (3.12, 3.13).

So far, so good. There is another issue related to Combescot and Pogosov Eqs. (12.4) - (12.5) from paper 1 where they discuss so-called “transfer assisted exchange” scattering. Here the intrigue is formulated as

Due to this “transfer assisted exchange” scattering, here also it is not possible to construct an effective bosonic Hamiltonian for Frenkel excitons which is hermitian, as physically required, and which produces the same matrix element as the one of the exact fermionic Hamiltonian.

and, even more strongly,

This is why it seems hard to guess physical results like the scattering rate of Frenkel excitons, the ground state energy of N Frenkel excitons or the nonlinear susceptibilities of materials having such excitons, without a precise calculation of these quantities using the procedure developed in this paper. These problems will be addressed in a near future.

Reading it one can indeed expect that everything will be different for Frenkel excitons. However, in their next paper [3 — Pogosov, W. & Combescot, M. Ground state energy of N Frenkel excitons. Eur. Phys. J. B 68, 183–192 (2009) published back to back with paper 1] the authors write (also pasted below as a picture)

So that, in the end, all the exchange processes appearing in 〈H〉N finally disappear, as if Frenkel excitons were true elementary particles, within terms in 1/Ns.

It makes me feel quite confident: the technique developed by Monique Combescot provides the expected results, although sometimes in very cumbersome fashion. Unfortunately, she does not compare her results with the results of previous works, but if I use her Eq. (4.23) of paper 3 and rewrite it ($N\gg1$, $\bar V(00;00) = 2 U_0/\mathcal V$, $\mathcal V = L^3$ is the volume) as $\langle H\rangle = N E_0 + N^2 U_0/\mathcal V$. This result, naturally, leads (at $T=0$) the pressure $P = - d\langle H\rangle/d\mathcal V = U_0(N/\mathcal V)^2$ and the “sound” (Bogolyubov) velocity $u = \sqrt{dP/d\rho}$, $\rho = m (N/\mathcal V)$ in full agreement with Eq. (20) of Agranovich and Toshich.

Differences with ideal weakly interacting bosons can and should occur in the next order in the exciton density (Combescot’s small parameter $N/N_s$, where $N_s$ is the number of atomic sites in the volume $\mathcal V$).

To put it in other words, I think that there is no fundamental issue at least with Frenkel excitons. I am not against complicated methods which support otherwise expected results (or derived previously in different fashion), moreover, it is helpful to see the limitations of the theory.

so the "disagreement", at least in this Frenkel case, boils down to who has the simple treatment, and who has the cumbersome one.

My reply to Misha on this aspect is as follows:

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Dear Misha,

Thank you for this interesting feedback. Meanwhile, I followed your beautiful trip in the mountains at a distance thanks to the combined generosity of yourself and the Internet, that both grace us with material worthy of National Geographic! What a sensation of depth, immensity and distances. I had the feeling of looking directly at the eleven time zones of Russia in your pictures of those mountains straddling over unending forests.

Regarding your comments on Combescot. As I understand, as far as Frenkel excitons are concerned, the disagreements vanish into perceptions of who owns the complexity. She writes «[Agranovich and coworkers] turns out to be rather complicated technically, in contrast to our method which is quite straightforward» while you write «the technique developed by Monique Combescot provides the expected results, although sometimes in very cumbersome fashion» or «I am not against complicated methods». I should have my own opinion about whose method is simpler and whose more cumbersome but unfortunately my lack of practice doesn't let me judge. That is still an interesting divide between the two "schools" and, more importantly, it doesn't result in disagreement in the results, which was my initial concern (the problem of validity doesn't pause itself in quite the same terms if everybody agrees).

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Being a practical and efficient person, he concludes:

I would like to suggest the following procedure. Let us indeed identify the system of interest, e.g., magnons or Frenkel excitons (to begin with) and try to “attack” it. I would suggest that you find a weak point in my analysis above (I can well omit something important, my analysis above is quite superficial, of course), or you can find weak arguments in bosonization of magnons. The only thing I would like to ask you is to avoid statements like “Combescot claims, etc.”: I would prefer to discuss physics with you rather than with argue with Combescot arguments (as we interpret these arguments).

and this is my reply:

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You are entirely correct that progress at this stage requires to find a clear point that can be described as problematic one way or another, in a more substantial form than our interpretation of someone's claims.

But none of the two candidates of your 13th of August email—which are magnons and Frenkel excitons—survive your analysis. You make your point compellingly there. Magnons, I agree, are surely not problematic regarding the composite-particle aspects and the Shiva formalism would presumably provide [modulo this being simpler or more cumbersome depending on tastes] the same results. More surprisingly to me is that Frenkel excitons—which bring about compositeness—also appear not to be problematic after all, as per your present discussion, despite her claiming that «they definitely differ from elementary bosons by various means which makes all intuitive handling of these excitons rather dangerous». If I followed your line of thought, it all remains buried within the formalism, with no concrete or at least severe consequences. I will make a more careful reading of her Frenkel case, including the follow-up works you highlight, to try to condense this into a short statement I'd be happy to write in Microcavities, along the lines of "this stays at the level of the formalism". Doing so, I will see if it is within my reach to connect this back to Haldane, since the original question was whether this had any connection to this 1D exact mapping method. This remains unclear to me...

The immediate (pending) conclusion of our discussion is then that there is no serious contention in the cases at hand and that claims of fundamental breakdown of particle interactions when involving compositeness are oversold, or at least are not obvious and flagrant in general.

The pushback would bring us back to Wannier excitons, as in this case I believe she has claims of strong qualitative departures (the localisation of Frenkel excitons seems to settle the basic issue in the terms described above, and this seems correct: compositeness + delocalisation is problematic, not compositeness alone). Even in her Frenkel paper, however, she highlights that «the replacement of Wannier excitons by elementary bosons misses the correct detuning behavior of all nonlinear optical effects induced by nonabsorbed photons.» In this case, she names an observable (third-order nonlinear susceptibility $\chi^{(3)}$) which is qualitatively different from the coboson formalism $\delta^{-3}$ as opposed to effective hamiltonians $\delta^{-4}$. I know she also has other, similar claims on Faraday rotation, Stark effects and dark exciton condensates... but that's Wannier, not Frenkel. So possibly an entirely different question from the original one. Still, let me ask you, out of curiosity, what's your brief take on her detuning case regarding exciton interactions, where Coulomb becomes negligible but indistinguishability and exchange remain structural.

Let me conclude by assuring you that I'm not dodging the issue; you have convinced me that there is likely no issue in such cases that were the most natural candidates for a bridge with orthodox (Haldane) bosonization. The point is that I'm not taking sides in the argument, I'm not defending Combescot, just trying to understand how valid her argument could be. So this brings me to concede that there is little more for me to do than properly review her literature for the Frenkel case, fully confirm your reading, and pass the hot potato to the Wannier case. I hope this doesn't read like circling around the problem.

Best regards,

Fabrice.

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05:56 (CET).0Ccip (AF).

19 September (2026)

I'm coming back to Jacob's observation that one-collapse distributions are very noisy.

For small enough $N$, we can get the exact $\binom Nk$ distribution; one question is how this deviates from the ensemble average. This is the sort of problem we studied in depth for observables, now asked for the distribution as a whole.

The other natural question is, if $N\gg 10^3$, say, it becomes impossible to sample everything. Then how does a sub-sample fluctuates around the ideal distribution?

All those are natural questions and I take them in part, starting with Jacob's main concern of how noisy one collapse distribution gets as compared to its ensemble average counterpart.

Doing so I realized that there is a general decomposition for $D_\alpha(d)$ the distribution of distances for a given $C_\alpha$: \begin{equation} \label{eq:0BRlx} E\bigl[D_2(d\,|\,\mathcal{C})\,\big|\,\alpha\bigr]=D_\alpha(d)=D_{\rm ind}(d)+C_\alpha K(d), \end{equation} with $$D_{\rm ind}\equiv\tfrac{d}{16}(8+d^4)e^{-d^2/2}$$ is the law of two photons on the donut and $$K\equiv\tfrac d8(8-8d^2+d^4)e^{-d^2/2}$$ is the deviation around them, same type but weighted by $C_\alpha$ and such that $\int K=0$.

Anyway, we first pause on the first aspect of the problem, how does $D_2(d\mid\mathcal{C})$ fluctuates around $D_\alpha(d)$ as compared to ensemble averages at the same photon cost.

17:32 (CET).0BSb2 (AF).

17 September (2026)

The singles, with self-alignment and $N=2$ post-selection. We've been circling this for ages and I've never seen this so neatly discriminated. That $\ket{1,1}$ looks like my quantum dipole that I've been feeling is there somewhere. What's not clear is what do we see here that we didn't see months ago?

And this is the $M_2$ (in Daniel's notation) transformation. And he found out a $M_1$ one that pieces those branches together. The uncorrelated becomes a half-donut. What of the others?

23:12 (CET).0BHTl (AF).

16 September (2026)

A balanced thermal frame has three confounding factors, $(S,a,\psi)$:

• The phase $\psi$, uniformly distributed.
• The intensity $S=I_a+I_b$, $\Gamma(2)$-distributed.
• The shape $a=\sqrt{I_aI_b}/S=\sqrt{t(1-t)}$ with $t$ uniformly distributed (where $C_\alpha=a^2$).

The three are independent, so each can be removed separately:

• The phase by rotating every frame by its own $\psi$ (the oracle alignment) or self-alignment at large particle numbers.
• The intensity by drawing $N$ Poissonian at a fixed mean ($R\to1$, $R^{(3)}\to1$).
• The shape by fixing the same contrast for all frames.

This makes for $2^3=8$ combinations, that are shown in Fig. (0A3Ik):

Top row allows for intensity fluctuations, bottom row doesn't. Those fluctuations are those of the thermal state. If we don't allow them, it means each coherent state has the same mean intensity $S=|\alpha_a|^2+|\alpha_b|^2$, but each frame sampled from them will thus still fluctuate, according to the Poisson law (of coherent states). If we further fix $N$ itself (as opposed to the mean $\langle N\rangle$) then we have a $1-1/N$ term (as opposed to $1$ for Poisson fluctuations).

(a) removes nothing so is the same as Fig. (0A6pq)'s thermal. (b) removes the shape fluctuations (one $C_\alpha$) so only the phase + intensity vary. This is essentially identical to (a), with a slight decrease (e.g., 1.963 autocorrelation instead of 2). (c) removes the phase fluctuations and so only the shape + intensity vary, and this is the interesting thing: correlations change significantly both in shape and in magnitude. (d) keep only intensity fluctuations, and we have a constant $g^{(2)}$.

Something similar happens with $g^{(3)}$.

The following decomposition of the auto-correlation can be made: \begin{equation}\label{eq:0AHRO} g^{(2)}(\theta,\theta)=1+\underbrace{(R-1)}_{\text{intensity}}+ \begin{cases} R\,[2m_1^2+2\operatorname{Var}a] & \text{phase random},\\[2pt] 4R\operatorname{Var}a\,\dfrac{\cos^22\theta}{(1+2m_1\cos2\theta)^2} & \text{phase removed}, \end{cases} \end{equation} where $$\operatorname{Var}a=m_2-m_1^2=\frac16-\frac{\pi^2}{64}=0.012452 .$$ $R-1$ contributes the "intensity fluctuations" since it is the relative variance of the intensity from shot to shot, and nothing else:

$$R=\frac{\langle N(N-1)\rangle}{\langle N\rangle^2}=\frac{E_P[S^2]}{E_P[S]^2}=1+\frac{\operatorname{Var}S}{\langle S\rangle^2}$$

So $R-1=\tfrac12$ for two-mode thermal light ($S$ is $\Gamma(2)$), $1$ for single-mode thermal ($g^{(2)}(0)=2$, the Hanbury Brown–Twiss excess) and $0$ for any fixed $S$. It vanishes when the intensity is frozen and survives when phase and shape are frozen, i.e., the "intensity only" panel is flat at $1+(R-1)=R$.

We now provide the full two and three-angle correlations.

Defining $c_j\equiv\cos2\theta_j$, $m_k\equiv E[a^k]$ ($m_1=\pi/8$, $m_2=1/6$, $m_3=3\pi/128$ for the thermal spread; $m_k=m_1^k$ when the shape is frozen) and $R$, $R^{(3)}$ equal to $\tfrac32$, $3$ or to $1$, $1$: \begin{align} &\text{$\psi$ random:} \nonumber\\ & g^{(2)}=R\,[1+2m_2\cos2\Delta\theta], && g^{(3)}=R^{(3)}\Bigl[1+2m_2\textstyle\sum_{j<k}\cos2\Delta_{jk}\Bigr], \label{eq:0AVxE}\\[1em] &\text{$\psi$ removed:} \nonumber\\ & g^{(2)}=R\,\frac{1+2m_1(c_1+c_2)+4m_2c_1c_2}{(1+2m_1c_1)(1+2m_1c_2)}, && g^{(3)}=R^{(3)}\,\frac{1+2m_1e_1+4m_2e_2+8m_3e_3}{\prod_j(1+2m_1c_j)}, \label{eq:0ALSn} \end{align} where $e_1=c_1+c_2+c_3$, $e_2=c_1c_2+c_1c_3+c_2c_3$ and $e_3=c_1c_2c_3$ are the elementary symmetric functions of the $c_j$, so that $\prod_j(1+2ac_j)=1+2ae_1+4a^2e_2+8a^3e_3$.

21:26 (CET).0Akvm (AF).

15 September (2026)

I've spent some time trying to classify and decide what to retain of the various (and innumerable) possible spatial correlation functions we've been playing with in the last several months. Here's a brief summary.

The least arguable is Glauber's correlation function defined in the canonical way: \begin{equation}\label{eq:0AwdV} g^{(2)}(\theta_1,\theta_2)\equiv\dfrac{\langle{:}\hat n(\theta_1)\hat n(\theta_2){:}\rangle}{\langle\hat n(\theta_1)\rangle\langle\hat n(\theta_2)\rangle}\,. \end{equation}

Then there are our definitions which differ principally by normalization. Daniel's definition: \begin{equation}\label{eq:0AgJI}\tilde g^{(2)}(\theta_1,\theta_2)\equiv{\tilde\Theta^{(2)}(\theta_1,\theta_2)\over\tilde\Theta^{(1)}(\theta_1)\tilde\Theta^{(1)}(\theta_2)}\,.\end{equation}

Note that $\tilde\Theta^{(2)}(\theta_1,\theta_2)\equiv\dfrac{\langle{:}\hat n(\theta_1)\hat n(\theta_2){:}\rangle}{\langle N(N-1)\rangle}$ is normalized so that $\iint\tilde\Theta^{(2)}(\theta_1,\theta_2)\,d\theta_1 d\theta_2=1$ and for isotropic cases, the denominator of Eq. \eqref{eq:0AgJI} is constant so that $\tilde g^{(2)}/(2\pi)^2$ is also a pdf. This makes it the joint law of the positions of a pair drawn at random among all the pairs.

Then our few-photon confounder-free correlation function, i.e., gauging-out the phase (self-alignment), post-selection to $N=2$ and fixing or not the quantum state ($C_\alpha$, thermal, Fock, etc.) This I will call $g^{(2)}_\sharp$ in agreement with notations from measure theory (pushforward distribution). This is defined on its canonical support $S=\{\operatorname{Im}Z=0,\ \operatorname{Re}Z>0\}$:

\begin{equation}\label{eq:0Av0D} g^{(N)}_\sharp\equiv(2\pi)^{N}\tilde\Theta^{(N)}\big|_S \end{equation} where $$Z\equiv\sum_je^{2i\theta_j}\,.$$

Operationally, $g^{(2)}_\sharp$ is coincidences over uniform chance, while Glauber's $g^{(2)}$ is coincidences over accidentals.

Then we have the variants of the above to particular situations, mainly, $g^{(2)}$ (standard Glauber) or $\tilde g^{(2)}$ (Daniel's pdf) applied to the frames once rotated, in which case I append a "al" (for "aligned") subscript.

Those cases are compiled in this figure:

One-photon observables. A coherent frame $\vert\alpha_a,\alpha_b\rangle$ is $N$ independent photons on the intensity $$I(r,\theta)=\frac{r^2}{\pi}e^{-r^2}\,[1+2a\cos(2\theta+\psi)],\qquad a=\frac{\sqrt{I_aI_b} }{I_a+I_b}\in[0,\tfrac{1}{2}],\qquad \psi=\arg(\alpha_a/\alpha_b)$$

Glauber $g^{(2)}$: $$g^{(2)}(\theta_1,\theta_2)=R\,[1+2\mathcal{C}\cos2\Delta\theta]$$ thermal $\tfrac{3}{2}+\tfrac{1}{2}\cos2\Delta\theta$, RPCS $1+\tfrac{1}{2}\cos2\Delta\theta$, $\vert1,1\rangle$ $\tfrac{1}{2}+\tfrac{1}{2}\cos2\Delta\theta$, uncorrelated $1$. Same-mode pairs ($aa$, $bb$) are flat, cross pairs ($ab$) go as $1+\cos2\Delta\theta$, and $2\mathcal{C}$ is the fraction of cross pairs, so the balanced states share the modulation $\tfrac{1}{2}\cos2\Delta\theta$ and differ by their constant part $R$.

If post-selecting exactly $N$ photons, then $$g^{(2)}(\theta_1,\theta_2)=(1-\tfrac{1}{N})[1+2\mathcal{C}_N\cos2\Delta\theta]$$ where $$\mathcal{C}_N\equiv\frac{\langle n_an_b\rangle_N}{\langle N(N-1)\rangle}=\frac{C_{11}^{(N)}}{C_{20}^{(N)}+2C_{11}^{(N)}+C_{02}^{(N)}},$$ so that, e.g., at $N=2$, thermal $\tfrac{1}{2}+\tfrac{1}{6}\cos2\Delta\theta$, RPCS $\tfrac{1}{2}+\tfrac{1}{4}\cos2\Delta\theta$, $\vert1,1\rangle$ $\tfrac{1}{2}+\tfrac{1}{2}\cos2\Delta\theta$, uncorrelated $\tfrac{1}{2}$.

Pair density (Daniel's pdf): $$\tilde\Theta^{(2)}(\theta_1,\theta_2)=\frac{\langle:\hat n(\theta_1)\hat n(\theta_2):\rangle}{\langle N(N-1)\rangle}=\frac{1+2\mathcal{C}\cos2\Delta\theta}{(2\pi)^2}$$ $$\tilde g^{(2)}(\theta_1,\theta_2)=\frac{\tilde\Theta^{(2)}(\theta_1,\theta_2)}{\tilde\Theta^{(1)}(\theta_1)\,\tilde\Theta^{(1)}(\theta_2)}=\frac{g^{(2)} }{R}=1+2\mathcal{C}\cos2\Delta\theta$$ ($\hat n(\theta)$ the angular photon-density operator; $\tilde\Theta^{(1)}=1/2\pi$ for a random phase). The visibility is $2\mathcal{C}$: thermal $1+\tfrac{1}{3}\cos2\Delta\theta$, RPCS $1+\tfrac{1}{2}\cos2\Delta\theta$, $\vert1,1\rangle$ $1+\cos2\Delta\theta$, uncorrelated $1$; unchanged by post-selection at fixed $N$ for balanced states.

Pushforward $g^{(N)}_\sharp$: For $N=2$, the support is the segment $\theta_1+\theta_2=0$, $\lvert\Delta\theta\rvert<\pi/2$, and \begin{equation}\label{eq:0Bo2R}g^{(2)}_\sharp(\Delta\theta)=1+2\mathcal{C}\cos2\Delta\theta=\tilde g^{(2)}=\frac{g^{(2)} }{R},\qquad p(\Delta\theta)=\frac{g^{(2)}_\sharp(\Delta\theta)}{\pi}\end{equation} is a probability density of the shape of a two-photon frame: thermal $1+\tfrac{1}{3}\cos2\Delta\theta$, RPCS $1+\tfrac{1}{2}\cos2\Delta\theta$, $\vert1,1\rangle$ $1+\cos2\Delta\theta$, uncorrelated $1$, all on $\lvert\Delta\theta\rvert<\pi/2$. For balanced states it is the same at every $N$ and equals the pooled $\tilde g^{(2)}$; for unbalanced thermal light ($\bar n_a\ne\bar n_b$) $\mathcal{C}_N$ depends on $N$ and it does not.

Aligned one-photon law:

First, the "oracle" alignment, i.e., each classical frame is rotated by its own true $\psi$: $$\tilde\Theta^{(1)}_\mathrm{al}(\theta)=\frac{1+2\bar a\cos2\theta}{2\pi}$$ RPCS: the dipole $(1+\cos2\theta)/2\pi$; thermal $(1+\tfrac{\pi}{4}\cos2\theta)/2\pi$; uncorrelated $1/2\pi$; $\vert1,1\rangle$: no oracle.

Second, self-alignment at $N=2$ gives instead, for every state, $\tilde\Theta^{(1)}_\mathrm{al}(\theta)=\tfrac{2}{\pi}[1+2\mathcal{C}\cos4\theta]$ on $\lvert\theta\rvert<\pi/4$, the marginal of the pair law.

Aligned Glauber $g^{(2)}_\mathrm{al}$, first, oracle: Glauber's function on the aligned frames, $R$ times the aligned pair density over the product of the aligned one-photon laws: $$g^{(2)}_\mathrm{al}(\theta_1,\theta_2)=R\left[1+4\operatorname{Var}a\,\frac{\cos2\theta_1\cos2\theta_2}{(1+2\bar a\cos2\theta_1)(1+2\bar a\cos2\theta_2)}\right]$$ Thermal: $$\tfrac{3}{2}+\left(1-\tfrac{3\pi^2}{32}\right){\cos2\theta_1\cos2\theta_2\over(1+\tfrac{\pi}{4}\cos2\theta_1)(1+\tfrac{\pi}{4}\cos2\theta_2)}\,,$$ i.e. $1.523$ on the axis ($\theta_1=\theta_2=0$), $1.5$ at $45^\circ$, $3.12$ at the waist ($\theta_1=\theta_2=\pi/2$), $1.305$ between axis and waist; the autocorrelation is $g^{(2)}_\mathrm{al}(\theta,\theta)=\tfrac{3}{2}+(1-\tfrac{3\pi^2}{32})\cos^2 2\theta/(1+\tfrac{\pi}{4}\cos2\theta)^2$. RPCS and uncorrelated: $1$ everywhere ($\operatorname{Var}a=0$). Dividing by $R$ gives the aligned normalized law $\tilde g^{(2)}_\mathrm{al}=1+4\operatorname{Var}a\cos2\theta_1\cos2\theta_2/[(1+2\bar a\cos2\theta_1)(1+2\bar a\cos2\theta_2)]$: thermal $1+(\tfrac{2}{3}-\tfrac{\pi^2}{16})\cos2\theta_1\cos2\theta_2/[\ldots]$, worth $1.016$, $1$, $2.08$, $0.87$ at the same four points.

Aligned Glauber $g^{(2)}_\mathrm{al}$, second, Self-aligned two-photon frames (exactly $N=2$, $R=\tfrac{1}{2}$). The aligned pairs sit on the anti-diagonal and the aligned one-photon law is the pair law's own marginal, so for every state $$g^{(2)}_\mathrm{al}(\theta_1,\theta_2)=\frac{\pi}{4}\,\frac{\delta(\theta_1+\theta_2)}{g^{(2)}_\sharp(\theta_1-\theta_2)},\qquad \tilde g^{(2)}_\mathrm{al}(\theta_1,\theta_2)=\frac{\pi}{2}\,\frac{\delta(\theta_1+\theta_2)}{g^{(2)}_\sharp(\theta_1-\theta_2)}:$$ the aligned Glauber function is the inverse of the shape distribution. Weight of the ridge, $\tfrac{\pi}{4}/g^{(2)}_\sharp(\Delta\theta)$: thermal $\tfrac{\pi}{4}/(1+\tfrac{1}{3}\cos2\Delta\theta)$, RPCS $\tfrac{\pi}{4}/(1+\tfrac{1}{2}\cos2\Delta\theta)$, $\vert1,1\rangle$ $\tfrac{\pi}{4}/(1+\cos2\Delta\theta)$ (divergent at $\Delta\theta\to\pm\pi/2$), uncorrelated $\tfrac{\pi}{4}$.

Self-aligned of other $N$ need be computed independently. The oracle-alignment is, in contrast, independent of $\langle N\rangle$; post-selecting exactly $N$ photons only replaces its prefactor $R$ by $1-\tfrac{1}{N}$.

Other important quantities that pin the physics:

The pair rate $$R\equiv{\langle N(N-1)\rangle\over\langle N\rangle^{2}}$$ with

• $R=\frac32$ thermal
• $R=1$ RPCS
• $R=1-\frac1N$ for frames post-selected with $N$ photons

The contrast $$\mathcal{C}\equiv{C_{11}\over\langle N(N-1)\rangle}=\frac{C_{11}}{C_{20}+2C_{11}+C_{02}}$$ which is

• $\mathcal{C}=\frac16$ for thermal,
• $\mathcal{C}=\frac14$ for RPCS,
• $\mathcal{C}=\frac12$ for $|1,1\rangle$.

This makes $2\mathcal{C}$ the fraction of pairs with one photon in each vortex. $\langle N(N-1)\rangle=R\langle N\rangle^{2}$ is the mean number of ordered pairs per frame. Note that $C_{20}+2C_{11}+C_{02}=\langle N(N-1)\rangle=\langle{:}\hat N^{2}{:}\rangle$.

Note that $C_{02}+C_{20}$ is the number of same-mode pairs. Since $\langle N(N-1)\rangle$ counts ordered pairs while $C_{11}$ counts the cross-mode pairs once (hence the 2 in front of it), $2\mathcal{C}$ is the fraction of cross-mode pairs, ordered or not. This rules the contrast of the interference.

The sector contrast $\mathcal{C}_N$ is the same quantity in the sector of exactly $N$ photons, $$\mathcal{C}_N\equiv\frac{\langle n_an_b\rangle_N}{N(N-1)}=\frac{C_{11}^{(N)}}{C_{20}^{(N)}+2C_{11}^{(N)}+C_{02}^{(N)}}\,,$$ with $N(N-1)=\langle N(N-1)\rangle_N$. It equals $\mathcal{C}$ for all the balanced states and depends on $N$ for unbalanced thermal light; the pooled contrast is the pair-weighted average of the sectors, $\mathcal{C}=\sum_NP(N)N(N-1)\mathcal{C}_N/\sum_NP(N)N(N-1)$.

16:54 (CET).0Akul (AF).

14 September (2026)

And the fixed $N$ thermal oracle for various $N$:

But this is less interesting than it looks, the dependence is simply: $$g^{(2)}_{N}(\theta_1,\theta_2)=\Bigl(1-\frac1N\Bigr)\frac23\;g^{(2)}_{\rm th}(\theta_1,\theta_2)$$ where $g^{(2)}_{\rm th}$ is panel (c) of Fig. (0AU6n). This happens to be, however, Daniel's $\tilde g^{(2)}(\theta_1,\theta_2)/[\tilde g^{(1)}(\theta_1)\tilde g^{(1)}(\theta_2)]$, so the intensity confounder is central to our definitions of correlations.

22:00 (CET).0AXPm (AF).

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And this to discuss the dominance (or not) of the phase as a confounding factor, along with a variety of other results:

The first result is how the correlation map looks on $[-\pi/2,3\pi/2]^2$.

Then the strong qualitative changes for the oracle if fixing (panel b) $N=2$ or not (panel c) $\langle N\rangle$ the number of particles. Not that the latter case is $N$-independent, but the former is not (will look at that next).

Panel (a) is the density of where the pairs land: in the bowtie, a lot of them in one lobe on the left, a lot on that on the right, one in each lobe, etc. Very few lobe-waist or wait-waist occurrences.

So a moot, trivial reading is that removing the phase as a confounder kills the correlations, because most of the pairs are now little correlated. That's the darkish four plateaux we see in panel (c), in the areas of the bright spots of panel (a).

But that our two-photon everywhere story: not quantity but quality. In fact we are distilling, re-routing, so to speak, photons to regions where they become more correlated. This should be studied more closely.

21:48 (CET).0AO3q (AF).

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I realize I have ill-defined the problem of stripping out the phase. From our Simpson paradox approach, the question is, if you remove the phase as the confounding factor, what does the correlation map become from the other confounding factors (shape through $C_\alpha$ and intensity)?

Mainly, we want to know whether the remaining correlations are still strong or just marginal, or maybe they are even stronger but redistributed, etc. We expect the RPCS to become uncorrelated in such a way.

That's an interesting question. What is not well-put, however, is "what do you mean exactly by removing the phase?" Now there is how we do it in ergoli (Natalia's paper): gauge it out. You can always do it. Makes $\sum_j e^{2i\theta_j}$ real. Fine. And that's what I had in mind. But from the thermal—or any classical—point of view, there is a more exact way: just pin the phase of the coherent state and only fluctuate their intensity imbalance. It would be equivalent as someone, let's say an "oracle", knowing the "true" phase. These two way to get rid of the phase are different, starting with the fact that Fock states don't have a "true" phase, so you can only ergolize them.

This is the oracle for thermal and RPCS, a better version of Fig. (09MmR) in particular with the good color code:

This is intensity independent, as should be.

Now the point is, for self-alignment (gauging out the phase), this is on the other hand strongly $N$ dependent. Since there is a normalization, even empty and single-photon frame will play a direct role. One recovers the oracle result when $N\gg1$, but at small $N$, it's a whole new universe:

Large $N$ recover the oracle result, fine... Small $N$ bring these weird geometric patterns, that come from the farfalle. This is more recognizable by shifting the axis, to unveil the four segments of perfect $N=2$ (not $\langle N\rangle=2$) correlations (farfalla):

The horizontal-vertical blue lines are single photons (frequent at such low intensities) falling on the axes by the self-alignment and thus killing the correlations there.

These intensity-dependent features show the necessity to fix $N=\mathrm{cste}$ at small $N$, and that's the farfalle story. We probably got there by intuition, luck or disgust of what fluctuating $N$ was giving us (we looked at that too, I remember).

What I remain unsure of is the practicality and physical meaning of those low-$N$, fluctuating, self-aligned results. I also need to find a way to make a subtle story simple. We could not refer to an oracle and assume high $N$, or invoke an oracle and be $N$-independent. But that's a lot to explain.

21:23 (CET).0AwvF (AF).

12 September (2026)

In the wake of Fano's argument, which sets phase as the cornerstone of the mechanism (it is even in the title), it is good to emphasize our (Salazar et al.)'s argument of confounding factors. First, Fano is explicitly on Fock states $\ket{1_a,1_b}$ although he discusses Hanbury Brown's thermal light. We do separate them quite strongly and attribute different mechanisms to both. Second, phase is one possible confounding factor. It is the the one for RPCS, since there is none other, but thermal states could get rid of their phase—that's precisely what we did with Natalia—and still be correlated due to other confounding factors.

Not aligned (phase random): \begin{equation} g^{(2)}(\theta_1,\theta_2)=\tfrac32\bigl[1+\tfrac13\cos2(\theta_1-\theta_2)\bigr] =\tfrac32+\tfrac12\cos2(\theta_1-\theta_2). \label{eq:09beI} \end{equation}

Aligned (phase pinned to zero): \begin{equation} \overleftrightarrow{g}^{(2)}(\theta_1,\theta_2)=\frac32 +\Bigl(1-\frac{3\pi^{2}}{32}\Bigr) \frac{\cos2\theta_1\cos2\theta_2} {\bigl(1+\frac\pi4\cos2\theta_1\bigr)\bigl(1+\frac\pi4\cos2\theta_2\bigr)}\,. \label{eq:09aKe} \end{equation}

From this, the autocorrelations read $$g^{(2)}(\theta)=2\,,\qquad \overleftrightarrow{g}^{(2)}(\theta)=\frac32+\Bigl(1-\frac{3\pi^{2}}{32}\Bigr) \frac{\cos^{2}2\theta}{\bigl(1+\frac\pi4\cos2\theta\bigr)^{2}}\,.$$ With gorgeous correlations:

In particular, this brings us to a new fundamental bunching constant: $$\boxed{g^{(2)}(\pi/2)=\frac{40-12 \pi }{(4-\pi)^2}}$$ which is approx $3.12254$.

If not the result of working late, we have this beautiful result that the phase as a confounding factor is not even central to the phenomenon, it merely smoothes out the correlations and distribute them equally around the rim. Intensity fluctuations are much more dominant, as they venture to a >3 suprathermal chaos into the waist, and, without phase whatsoever, bring "coalescence" to the level of RPCS in the lobes of the aligned dipoles (which is, I think, what we used to call the bow-tie with Daniel). Overall, their correlations remain on both sides of the non-aligned case $g^{(2)}=2$. Cross-correlations are also interesting, as they push the "forbidden" configurations (perpendicular) to some amount of bunching too. Nothing is suppressed!

I believe this shows convincingly that everybody has been missing a big part of the picture, conflating Fock and thermal cases, disregarding quantum states, being oblivious to the confounding factors, ensemble averages, etc.

23:33 (CET).092eR (AF).

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I should try to reproduce for my own benefit and as first step of a diagrammatic theory of frequency-resolved photon correlations, the Fano arguments on a full perturbation theory of the HBT which-way interpretation, which can be truncated to these two diagrams:

He says:

Variants of these diagrams, in which the vertices lie in different sequences from top to bottom, correspond to the appearance of different probability amplitudes, e.g., of $C_{ub}$ rather than $C_{av}$, in the system of equations of perturbation theory. Consideration of these individual variants is made unnecessary by two theorems, which can be proved by operator methods⁶ or verified by ordinary perturbation theory.⁷ The theorems are: (a) $C_{cd}(t)$ consists of the sum of two contributions corresponding to the two diagrams of Fig. 1. (b) Because each diagram consists of two unlinked fragments, its contribution is the product of two factors corresponding to the two fragments. Moreover, the four fragments of which the two diagrams consist have identical structures. Each of them represents the emission of a photon by an atom and its absorption by another atom. Therefore, the probability amplitude of this second-order process is the basic ingredient of our calculation.

where ⁶ are references (one Feynman) and ^7,8 are further precisions:

Clearly he did all this himself. Maybe less clearly, some assumptions, like 8, appear less solid than others, so although it seems a simple problem, there is a lot under the hood.

18:21 (CET).09NXB (AF).

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This animation shows both the disconnection of the "jumps" from the peaked structure of the distribution (this is just a moot change of sign magnified by the log-stretching) and the effect of sub-sampling, with only 1000 points. First, at criticality, where sampling is very good at seeing the object at small $k$—the particular realization we're dealing with. Interestingly, the sub-sampling brings us closer to the real ensemble average. If we ignore the specificities of the individual, we get a better view of the set it comes from:

In contrast, the donut struggles more at staying with its particular case, and falls back on features of the ensemble; also, it fluctuates typically beyond the single trajectory, as opposed to the dipole which fluctuates inside.

Those are features of self-averaging (donut) or its breaking (dipole), but they are visible at small $k$ only, since ergodicity breaking is a feature of $k/N$, not $k$ alone and as $k\to N$ they both converge to the particular collapse result. In all cases, the ensemble average (purple points) at the same photon cost can be seen to wreck down completely the underlying structure.

10:15 (CET).09tpC (AF).

11 September (2026)

Jacob complains that $\binom{10^3}2\approx 10^6/2$ is not enough data to have a smooth single-collapse $D_2(p\mid\mathcal{C})$, though he compares that with $3\times 10^6$ independent-sample distributions which are, indeed, smooth. I don't think I ever looked at the noise in such distributions, but should, now. I think the single collapse will still be better at the same photon cost. So I'm not too worried about this temporary failure.

More importantly and adversarially, there is a bias in the experimental data in the radius, which makes agreement with the theory less than ideal. It's not critical because we study angular correlations, but since we introduce the story through absolute distances in space—which involve the radius—that brings unwelcome complications at the stage of the week and the stage of my mood.

17:21 (CET).08Ssx (AF).

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Adam and Eve's law in statistics are actually fringe terminology, from Blitzstein et al. I was under the impression a real Adam had coined the law of total expectation but this seems too basic to even have had a discoverer, and that Eve came as a wink. It's actually the other way around. Eve came first on this occasion from a mnemonic device:

And what came off her rib on this occasion—although it comes first indeed as targetting a more fundamental moment—is introduced as such:

So, yes, they are witty, and I'll keep the terminology for fun, but won't keep it for the paper.

12:37 (CET).082Ap (AF).

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This is the effect of sub-sampling, with few ($10^4$) and quite a lot, yet still manageable ($10^7$) samples, from one collapse:

The lines are when sampling over all subsets: no statistical noise. This one is re-introduced if we overlook some possibilities. Noise is not a feature of experiments, it is a feature of partial views, truncated treatments, botched jobs. Experiments can be perfect (mathematically smooth).

In more details, the donut is much more prone to noise, while the critical case is much more resilient to it: it's good to have some individuality, it makes you less affected by superficiality.

11:13 (CET).08yf5 (AF).

8 September (2026)

In the top Fig. of 073Rh, $k\sigma(k)$ becomes constant at criticality, which is the feature of the ensemble average. Both are critical—it's derived from the geometry, not of the averaging—so that's fine.

But if we focus on the single-collapse (quantum) average, things get interesting too, and maybe that's the sharpest picture. We must stay along a ray—$\kappa$ fixed and $N$ growing—since that's where ergodicity breaks. How to keep "individuality" is obvious: take a huge collapse and sub-sample from it to make $N$ grows, until you recover everything. So now I'm in the business of sampling within sampling within sampling!

The single-collapse standard-deviation, $k$-corrected, reads: \begin{equation} k\,\mathrm{sd}_{\mathcal{C}}(p_{\mathcal{C}}) =\sqrt{\rho(\kappa)}\;k\sigma(k) \;\xrightarrow[N\to\infty]{\kappa\ \text{fixed}}\; \begin{cases} \sqrt{\rho(\kappa)}\times0.479A & C_\alpha=\tfrac14,\\[3pt] \propto N^{-3/2}\to0 & C_\alpha\neq\tfrac14, \end{cases} \label{eq:07hYC} \end{equation} since $\rho$ is a function of $\kappa$ alone and $k\sigma= (k\langle\delta\rangle)(\sigma/\langle\delta\rangle)$ is $A\times0.479$ at criticality but $\propto k^{-1}\times N^{-1/2}$ elsewhere. $A$ is a critical amplitude, large-k constant to which $k\langle\delta\rangle$ converges to, i.e., $k\langle\delta\rangle(k)\to A$.

This gives us something like this (not plotting the independent ensembles data):

What is remarkable, besides the intended flat single-collapse variance (but the ensemble also remains flat), is that now the individuality of the collapse is altered: it is not smooth anymore, it becomes fractal: each added photon to the whole object, needed to grow $N$ as $\kappa$ remains constant, brings a kink in the individuality. If you don't have the full picture from the start, but need to get it as you go along, disruptions can happen: they are smooth but clear and can go to all orders. I'm inventing a new type of physics... psychological or sociological physics: find interpretations to universal themes of existence, especially of interactions between people, in fundamental objects. If you had not added this one point in your life, you'd have a sweet, smooth follow-up. But this new point got there and it made you jump slightly off course.

Jokes aside, this is also very nice, but I'm not sure we'll be using it right away...

10:48 (CET).07dA4 (AF).

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What we'll try to capture for Natalia's data:

• The gain $G$ as function of $k$ and in particular at $k=2$ and $k=3$, where it varies the most for different $C_\alpha$ (Fig. 3a)
• The strip: $k\sigma$ constant across $k=30$–$700$ at criticality against falling by $>15$ (atom) and $\approx100$ (donut), from independent samplings (Fig. 2d).
• The qualitative behaviours of $n_{\rm eff}(k)$: it should be flat ($\approx 4.5$) at criticality. It should produce a dip to $\approx 2.5$ at $k\simeq100$ and then increase for the atom. It should increase with slope $1/10$ for the donut. The dip's position reads $k_c$ and hence $1-2\sqrt{C_\alpha}$ from one frame. (Fig. 3b)

The EB$(p_\mathcal{C})$ could go in Fig. 3a as well [or if too heavy, removed entirely].

We could also show fixed $k$-ergodicity for all states and consider the slopes, but this is less important.

09:32 (CET).07oEz (AF).

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The result of some obsessive thoughts on criticality. At fixed $k$, everything is ergodic. Increase $N$ and the collapse mean converges on the ensemble always. The slope at which $\mathrm{EB}(p_{\mathcal{C}})$ falls to zero is smaller at criticality but this is quantitative. I won't discuss here the ergodicity of the variance itself, although this is probably interesting in connection with frustrated systems.

Along a ray, however, criticality becomes non-ergodic. Fixing the fraction $\kappa\equiv k/N$ and letting both grow, sees the donut and the atom still converge to zero, $\mathrm{EB}\propto1/N$, but for $C_\alpha=1/4$, the parameter converges to a constant. For $\kappa=1$, this constant is the single-shot width $0.479^2$. The collapse mean $p_\mathcal{C}$ remains an unbiased estimator but its variance doesn't go to zero. It still does for non critical cases, i.e., $\operatorname{Var}^2_{\mathcal{C}}(p_{\mathcal{C}})/\langle\delta\rangle\to0$ (roughly $N^{-1/2}$) at $C_\alpha\neq1/4$, but it goes $\to\sqrt{\mathrm{EB}_\infty(\kappa)}$ at $C_\alpha=1/4.$

The main picture that captures the bulk of the phenomenology is this one:

where the $y$-axis is $k(p-\langle p\rangle)$, i.e., $k(\langle\delta\rangle-\delta)$. The factor $k$ makes the standard deviation flat for the ensemble average and thus makes a neat divide between self-averaging, which closes the funnel, and non-self-averaging, that remains an opened fringe, within which trajectories move freely, being thus more representative of themselves than of the collective (although still unbiased estimators).

The mechanism is maybe best capture by $n_\mathrm{eff}\equiv(\langle\delta\rangle/\sigma)^2$ the number of effective chords (segments that make up the deficit) needed to provide the result. And there is a clean result for those, namely: $$\frac{n_{\rm eff}}{k}\to\frac{8(1-4C_\alpha)^{3/2}}{80+1536C_\alpha+624C_\alpha^2}$$ i.e., $n_{\rm eff}=k/10$ for the donut (exactly, the 10 being derived, not fitted), $n_{\rm eff}\approx k/229.5$ for the atom (so $\sigma/\langle\delta\rangle\to\sqrt{229.5/k}$) and, crucially, vanishes at criticality $C_\alpha=1/4$, so that the next term takes over, which is the constant $n_{\rm eff}=4.36$ (criticality). A fixed number of photons is enough to provide the details of a particular collapse. Adding more photons doesn't bring more information that would be information from the family itself from which the particular case is sample: the system is not self-averaging anymore.

This sharp behaviour is caused by the exact zero on the rim for the dipole. One can extend this behaviour to all states if considering the effect not of a zero, but of the depletion (minimum) there $\varepsilon\equiv1-2\sqrt{C_\alpha}$. Then every state is critical up to $k_c\equiv(2\pi/\sqrt2)\,\varepsilon^{-3/2}$, which is where one photon, on average, first lands in the depleted area, thus starting to "fill-up the criticality."

The skewness and kurtosis of the distributions are also interesting, as they differ for critical and non-critical cases (criticality being much more Gaussian, apparently). But I leave this out for now.

09:24 (CET).073Rh (AF).

2 September (2026)

Regarding the first question of 05MZV, he confirms that any two LG$_{\pm1}$ quantum states $C_\alpha$ and $C_\beta$ on the two-photon farfalle are rescaled versions of the same distribution: $${g_\alpha^{(2)}-1\over g_\beta^{(2)}-1}={C_\alpha\over C_\beta}\,.$$ So there is only one distribution, which can be stretched more or less. Or not at all, if $C_\alpha=0$ (not correlated), then $g_\alpha^{(2)}=1$. I believe this should be general, i.e., apply to $g^{(n)}$ on its $(n-1)$-dimensional farfalle.

Taking the most correlated case as a reference:

$$g_\alpha^{(2)}(\Delta\theta)=1+2C_\alpha\cos(4\ell\Delta\theta)$$

$C_\alpha={1\over4}$ is the critical value over which the state becomes quantum. He checked with two-mode squeezed vacuum, for which he finds:

$$g_\xi^{(2)}(\Delta\theta)=1+2{\bar n+{1\over2}\over2\bar n+{1\over2}}\cos(4\ell\Delta\theta)$$ with min $g^{(2)}$: $$\min g_\xi^{(2)}={2\bar n \over4\bar n+1}>0\,.$$

That settles nicely that the distribution touching impossibility is a particular case of quantumness: it implies it, but this is not necessary. That makes the contraposite of my theorem a bit moot: reduced to its extreme.

The proof is general an relies on cancellation of anomalous correlators through the phase re-alignment.

It's quite exciting because also the (related) second question gets answered positively. HG structures, once re-aligned, provide the same result: there is only one support and only one distribution on it. It only needs to be generalized (at least to $N>2$).

23:16 (CET).05xAK (AF).

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I initially made a statement on the different self-averaging of the variance (I retracted it now, not to confuse myself later, as I was incorrectly stating that variance doesn't self average). I was commenting that self-averaging properties of the variance could be notable and maybe related to spin ice and observables in strongly-correlated condensed matter, where fluctuations are the main concern. This still applies.

I had produced this beautiful figure, for fixed $N$ (hence not telling much about self-averaging):

The figure is correct but its interpretation wasn't: variance does self average.

I don't think I'll look much into the variance as the self-averaging of rays is enough for now. But still, for the record (and later comeback to this), every width self-averages only as $1/N$, while the donut's perimeter manages $1/N^2$ so there is some tension between them (which maybe is captured by self averaging of the mean, though).

14:32 (CET).05qyk (AF).

1 September (2026)

And now for the big one of today: we introduce the ergodicity-breaking parameter, the relative scatter—across collapses—of the quantity estimated from one of them: \begin{equation}\label{eq:054CV}\mathrm{EB}=\mathrm{Var}(\overline{\delta^2})/\langle\overline{\delta^2} \rangle^2\,.\end{equation} This comes from standard the time-averaged ergodic theory. $\mathrm{EB}\to0$ is ergodic while $\mathrm{EB}\to$ a non-zero constant is broken ergodicity.

Transcribed to multiphotonics, and using the deficit $\delta$ again as well as its variance: \begin{equation} \mathrm{EB}(p_{\mathcal{C}}) =\frac{\mathrm{Var}_{\mathcal{C}}(p_{\mathcal{C}})}{\langle\delta\rangle^2} =R_N, \qquad \mathrm{EB}(v_{\mathcal{C}}) =\frac{\mathrm{Var}_{\mathcal{C}}(v_{\mathcal{C}})}{E[v_{\mathcal{C}}]^2}, \label{eq:EB} \end{equation} Note that the second looks closer to the original tracking parameter, both being relative scatters of a second moment measured from one realisation.

As was the case for the related self-averaging, we have ergodicity at fixed $k$, as a function of $N$. But criticality breaks it for $k/N$, and the manifestation of this is a beautiful ladder: a factor $10$ increase of $N$ (and thus of $k$) results in a factor 10 decrease of the EB for non-critical cases. The critical case, however, remains essentially exactly constant:

21:10 (CET).05OrC (AF).

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An interlude to collect exact results regarding perimeters $\langle p\rangle$ and deficits $\langle\delta\rangle$ of $k$-gons on the unit circle. Jacob was wondering about them. They are not needed to compute the exact deficit since this is obtained as an average of differences from chords to arcs, but, in principle they are needed for the ensemble-average limits. And it's nice to have them.

By definition, $\langle\delta\rangle=2\pi-\langle p\rangle$.

The cases $k=2$ and $k=3$ are accessible for all states $C_\alpha$: $$ \langle p\rangle=\frac{4k}{\pi}\Bigl(1-\frac{2C_\alpha}{15}\Bigr),$$

The donut $C_\alpha=0$ is also available for all $k$ as a series: $$\langle\delta\rangle_{\rm donut}(k) =\sum_{n\ge1}\frac{(-1)^{n+1}\,2\,\pi^{2n+1}} {(k+1)(k+2)\cdots(k+2n)}\,,\ \qquad k\ge2 $$ The leading term is $\langle\delta\rangle\simeq2\pi^3/[(k+1)(k+2)]$.

Off the donut, there are no exact results for $k\ge4$ but asymptotics are interesting: \begin{align} \langle\delta\rangle&\simeq\frac{2\pi^3}{k^2(1-4C_\alpha)^{3/2}}\,, \quad(C_\alpha<1/4), \\ \langle\delta\rangle&\simeq\frac{A}{k},\ \ A=5.698\,, \quad(C_\alpha=1/4). \label{eq:asympt} \end{align}

We see again criticality behaving qualitatively differently.

19:48 (CET).05EKH (AF).

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This is the closed-form solution of an integral which popped up in Daniel's latest computation of the volumes of $N=4$ and 5 farfadelles. He could compute everything exactly but this one:

$$K=\int_0^{\pi/2} x\cot x\,\ln(1+2\sin x)\,dx\,.$$

Claude (Fable), however, could compute it:

\begin{equation}\label{eq:05d6l}\begin{split}K=\frac{35\pi^3}{288}&+\frac{\pi}{4}\ln^2 2+\frac{\pi}{8}\ln^2(2+\sqrt3)+\frac{\ln(2+\sqrt3)}{2}\operatorname{Cl}_2\!\left(\frac{\pi}{3}\right)+\frac{\pi}{2}\operatorname{Li}_2\!\left(-\frac12\right)\\&-2\,\mathrm{Im}\!\left[\operatorname{Li}_3\!\left(\frac{1+i\tan\frac{\pi}{12}}{2}\right)+\operatorname{Li}_3\!\left(\frac{1+i\tan\frac{5\pi}{12}}{2}\right)\right]\end{split}\end{equation} with $\operatorname{Cl}_2$ the Clausen function and Li$_s$ the polylogarithm (so we have dilogarithms and trilogarithms here).

That makes $$K=0.7315710425072642965\ldots$$ which is what I remember from his numerical evaluation of it. It does rely on special functions—which always look like a cheat—but at least that should exclude other nicer functions.

Claude (Fable 5):

The route: differentiating under the integral sign with respect to $a$ in $\ln(1+2a\sin x)$ makes the integrand collapse (since $\cot x\sin x=\cos x$), and the classical $\int_0^{\pi/2}\frac{dx}{1+b\sin x}=\frac{\arccos b}{\sqrt{1-b^2} }$ reduces everything to a one-dimensional integral, dispatched with $\int_0^{\pi/2}\frac{\theta\,d\theta}{\sin\theta}=2G$ and Ramanujan's $\operatorname{Ti}_2(2-\sqrt3)=\frac{2G}{3}+\frac{\pi}{12}\ln(2-\sqrt3)$. Catalan's constant then cancels entirely from the final result. Note that $\tan\frac{\pi}{12}=2-\sqrt3$ and $\tan\frac{5\pi}{12}=2+\sqrt3$ are the roots of $t^2-4t+1$, so the two trilogarithm arguments multiply to exactly $i$; these two constants appear to be genuinely irreducible (nothing simpler at 380 digits of PSLQ). The closed form is verified against direct numerical integration to 340 decimal digits.

Not very enlightening but that settles it.

19:35 (CET).05x8Y (AF).

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Self-averaging is the last piece I need to assemble Natalia's paper. This is a property of quenched disordered systems, in our case, brought by the collapse. A quantity computed from one such frozen collapse is self-averaging if its collapse-to-collapse relative fluctuation vanishes as the sample grows, $$ R_N=\frac{\mathrm{Var}(X_N)}{E[X_N]^2}\longrightarrow0 \qquad(N\to\infty),$$ and strongly so if $R_N\propto1/N$.

This is the condition for one large-enough sample to represent the ensemble average (ergodicity), and repeating the experiment tells us nothing we didn't already have. When it fails, every sample gives a different picture and only an ensemble means anything. Right our Natalia's ergodicity business!

But now to make it precise and quantify when it holds or not.

For the perimeter itself, we trivially have self-averaging from Eq. (1) at fixed $k$ and large $N$, $$\mathrm{Var}_{\mathcal{C}}(p_{\mathcal{C}})\simeq\frac{k^2\zeta_1}{N} \quad(\zeta_1>0), \qquad\qquad \mathrm{Var}_{\mathcal{C}}(p_{\mathcal{C}})\simeq \frac{k^2(k-1)^2\zeta_2}{2N^2}\quad(\zeta_1=0),$$ so $p_{\mathcal{C}}$ is strongly self-averaging off the donut and super-self-averaging at it—the standard deviation falling as $N^{-1}$ instead of $N^{-1/2}$. The only notable thing here is that the donut is much more self-averaging than the rest: this comes from its $\zeta_1=0$, i.e., rotation invariance, which says that no single photon carries information about the perimeter.

More than the perimeter, the deficit to that of the circle is more interesting, because instead of $2\pi$, the denominator becomes an increasingly smaller quantity.

The obvious self-averaging at fixed $k$ is not the most interesting: every $C_\alpha$ self-averages because adding photons to a collapse averages a fixed-size observable over ever more subsets. So we find self-averaging, although, again, with a different slope (super self-averaging for donuts):

It is more insightful to consider the self-averaging properties of, say, the biggest subset $\binom Nk$ which is when $k=\lfloor N/2\rfloor$. Or let's even generalize to any ray $k\propto N$. Then we find a breakdown of self-averaging for critical cases ($C_\alpha=1/4$):

17:53 (CET).05Krn (AF).