A CUE critical-point sum rule and the small-exponent Jensen bridge

Definitions and mathematical statement

Ashkan Nikeghbali's source question asks about the distribution of zeros of the derivative of a random unitary characteristic polynomial, especially near the unit circle, and about ${\mathbb{E}}[|f'|^s]$ as $s\to0$. The source does not specify the unitary ensemble, evaluation point, matrix-size limit, or normalization of $f$. Its extraction is marked reconstructed and unverified. The following precise formulation is the accompanying aid's CUE interpretation, not an implicit completion of all missing source data.

Let $N\geq2$, let $U_N$ be Haar distributed on the unitary group $U(N)$, and define \[ \Lambda_N(z)=\det(I-zU_N^*). \] Let $c_{1,N},\ldots,c_{N-1,N}$ be zeros of $\Lambda_N'$, counted with multiplicity. The aid reports the exact identity \[ \prod_{j=1}^{N-1}|c_{j,N}|=\frac{|\operatorname{Tr}U_N|}{N}. \] Almost surely these critical points lie in the open unit disk and none equals zero. Their total logarithmic radial depth is therefore \[ D_N=\sum_{j=1}^{N-1}\log\frac1{|c_{j,N}|} =\log N-\log|\operatorname{Tr}U_N|. \] The reported limiting law is \[ D_N-\log N\ \Longrightarrow\ W,\qquad \Pr(W\leq x)=\exp(-e^{-2x})\quad(x\in{\mathbb{R}}),\quad N\to\infty. \] Writing $K_N(r)=\#\{j:|c_{j,N}|\leq r\}$, the deterministic bound supplied by the aid is \[ K_N(r)\leq\frac{D_N}{\log(1/r)},\qquad 0<r<1. \] This controls critical points deep inside the disk but is not a complete near-boundary point-process law.

For the small-exponent bridge, fix $N$ and $0\leq r<1$, put $X_{N,r}=\log|\Lambda_N'(r)|$, and impose two-sided exponential integrability, for example ${\mathbb{E}}\exp(\delta|X_{N,r}|)<\infty$ and ${\mathbb{E}}\exp(\delta|X_{N,0}|)<\infty$ for some $\delta>0$. Under these explicit hypotheses the aid's Jensen interpretation takes the form \[ \left.\frac{d}{dq}\right|_{q=0} \left(\log{\mathbb{E}}|\Lambda_N'(r)|^q-\log{\mathbb{E}}|\Lambda_N'(0)|^q\right) ={\mathbb{E}} X_{N,r}-{\mathbb{E}} X_{N,0} =\int_0^r\frac{{\mathbb{E}} K_N(t)}{t}\,dt. \] The integral is zero at $r=0$. Rotational invariance supplies the connection between a fixed radial point and the angular Jensen mean. This formulation makes integrability an assumption; the supplied record does not establish it uniformly in $N$ or as $r\uparrow1$. The summary also reports an expansion of a Simm--Wei fixed-interior limiting formula giving two log-cumulants, but includes neither that formula, its normalization and limiting hypotheses, nor the cumulant values. These cannot be reconstructed as a precise additional theorem from this record.

Short English statement

For Haar unitary matrices, the supplied aid relates the product of critical-point radii exactly to the trace, yielding a limiting law for total radial depth and a bound on deep critical points. A conditional Jensen formula links the derivative of small-exponent moments at zero to the expected radial count. These statements do not specify the full boundary distribution requested in the original question.

Sources

• Local UnsolvedMath registry, permanent ID $20000628$, source code catalogue entry, supplied reconstructed source text and research summary; canonical local record aim-analysis-notes.json, notes[159].
• Ashkan Nikeghbali, Question 3, AIM workshop Random analytic functions, in Open Problems at the Random Analytic functions Workshop at AIM, Swaminathan Sethuraman, April 17, 2006, as identified by the supplied verification note: https://aimath.org/WWN/randomzeros/arcc1.pdf. This is a supplied citation only; no PDF was retrieved or compiled.
• The research summary names a Simm--Wei fixed-interior limit without a full bibliographic reference or its formula. The original literature field is empty.
• Research attempt 1: partial result; problem partially solved at the recorded run; supplied rigor label "proved", novelty confidence "low". The aid is machine generated, not peer reviewed, and must be independently verified before citation. Its novelty suggestion is not a verified priority claim.