--- title: Finite-Variance Extremality Conjecture url: https://www.emergentmind.com/topics/finite-variance-extremality-conjecture type: topic --- # Finite-Variance Extremality Conjecture Searching arXiv for the cited papers and related terminology to ground the article in current arXiv records. The expression *Finite-Variance Extremality Conjecture* does not designate a single universally fixed statement in the arXiv literature represented here. Instead, it appears as a family of extremality principles in which variance, a second-moment proxy, or a variance-type criterion governs the optimality of a probability law, a convex-geometric measure, a Gibbs phase, or a coupon-probability vector. In the materials considered here, the phrase is used for four distinct but structurally related problems: lower bounds for central and one-sided probability mass in relation to mean and variance, proved within the Gamma family [2303.17487]; the variance conjecture for centered log-concave measures and projections of the cube [1703.09973]; variance-based extremality criteria for translation-invariant phases of a three-state SOS model on the binary tree [1411.5886]; and the Doumas–Papanicolaou conjecture for the double Dixie cup problem, proved exactly by Long [2604.25108]. ## 1. Scope of the term and its principal formulations The formulations appearing under this label in the present corpus differ in ambient category, target functional, and extremizer. What they share is an extremal comparison driven by variance or a finite-variance surrogate. | Setting | Object | Extremal statement | |---|---|---| | Gamma laws | \(X_{\alpha,\beta}\) with \(E[X_{\alpha,\beta}]=\alpha\beta\), \(\mathrm{Var}(X_{\alpha,\beta})=\alpha\beta^2\) | \(P\{|X-E[X]|\le \sqrt{\mathrm{Var}(X)}\}\ge P\{|Z|\le 1\}\), and \(P\{X\le E[X]\}\ge \tfrac12\), within the Gamma family | | Convex geometry | Centered log-concave \(\mu\) on \(\mathbb{R}^n\) | \(\mathrm{Var}_\mu |x|^2 \le C\,\lambda_\mu^2 \int |x|^2\,d\mu(x)\) | | Tree spin systems | Translation-invariant splitting Gibbs measures | Extremality/non-extremality detected by variance-type or second-moment criteria | | Double Dixie cup | Completion time \(T_m(N)\) under coupon law \(p\) | \(\mathrm{Var}_p[T_m(N)]\ge \mathrm{Var}_u[T_m(N)]\), equality iff \(p=u\) | This suggests a common extremal template: a class of objects is equipped with a variance-sensitive functional, and a distinguished reference object—typically Gaussian, isotropic, or uniform—is conjectured or shown to optimize that functional. The distinctions are substantive, however. In the Gamma setting, the conjectural comparison is with the standard normal and concerns probability mass inside one standard deviation; in convex geometry it is an upper bound on \(\mathrm{Var}|x|^2\); in the SOS model it concerns phase extremality on a tree; and in the double Dixie cup problem it is a finite-\(N\) variance minimization statement. ## 2. Gamma-family extremality and Gaussian lower bounds For \(\alpha,\beta>0\), let \(X_{\alpha,\beta}\) be a Gamma random variable with density \[ f_{\alpha,\beta}(x)=\frac{x^{\alpha-1}e^{-x/\beta}}{\Gamma(\alpha)\beta^\alpha}, \qquad x>0. \] Its mean and variance are \[ E[X_{\alpha,\beta}]=\alpha\beta,\qquad \mathrm{Var}(X_{\alpha,\beta})=\alpha\beta^2. \] The two probability functionals studied are \[ P\{X_{\alpha,\beta}\le E[X_{\alpha,\beta}]\} \] and \[ P\{|X_{\alpha,\beta}-E[X_{\alpha,\beta}]|\le \sqrt{\mathrm{Var}(X_{\alpha,\beta})}\}. \] A scaling reduction collapses both functionals to a one-parameter problem. Writing \[ g(\alpha,\beta)=P\{X_{\alpha,\beta}\le \alpha\beta\}, \] the change of variable \(y=x/\beta\) shows \(g(\alpha,\beta)=g(\alpha,1)\), so it suffices to study \[ h(\alpha)=P\{X_{\alpha,1}\le \alpha\}=\frac{1}{\Gamma(\alpha)}\int_0^\alpha y^{\alpha-1}e^{-y}\,dy. \] Sun–Hu–Sun prove that for all \(\alpha>0\), \(h(\alpha)>\tfrac12\), and \[ \inf_{\alpha>0}h(\alpha)=\frac12. \] The proof outline has three components: \(h(\alpha+1)0\); \(\lim_{\alpha\to\infty}h(\alpha)=\tfrac12\) by a classical Gaussian-CLT argument for the sum of \(\alpha\) independent Exponential(1) variables; and \(h(\alpha)\to 1\) as \(\alpha\to 0+\) [2303.17487]. The two-sided analogue is \[ s(\alpha,\beta)=P\{|X_{\alpha,\beta}-\alpha\beta|\le \sqrt{\alpha}\,\beta\}=s(\alpha,1), \] so one defines \[ t(\alpha)=P\{|X_{\alpha,1}-\alpha|\le \sqrt{\alpha}\} =\frac{1}{\Gamma(\alpha)}\int_{\max(0,\alpha-\sqrt{\alpha})}^{\alpha+\sqrt{\alpha}} y^{\alpha-1}e^{-y}\,dy. \] The theorem states that for all \(\alpha>0\), \[ t(\alpha)>P\{|Z|\le 1\}\approx 0.6826895, \] and \[ \inf_{\alpha>0} t(\alpha)=P\{|Z|\le 1\}. \] Again, the key steps are strict monotonicity \(t(\alpha+1)0\) such that for every centered log-concave \(\mu\), \[ \mathrm{Var}_\mu |x|^2\le C\,\lambda_\mu^2\int |x|^2\,d\mu(x). \] If \(\mu\) is in isotropic position, so that \(E_\mu x=0\) and \(\mathrm{Cov}(\mu)=\mathrm{Id}\), then \(\lambda_\mu^2=1\) and the conjecture predicts \[ \mathrm{Var}_\mu |x|^2\le C\,n. \] The paper on projections of the cube proves this conjecture for a substantial family. Let \(B_\infty^n=[-1,1]^n\), and for \(E\in G_{n,n-k}\) let \(K=P_EB_\infty^n\) with \(p\) the uniform probability on \(K\). Theorem 1.1 states that there is an absolute constant \(C\) such that whenever \(1\le k\le \sqrt n\) and \(E\in G_{n,n-k}\), the measure \(p\) on \(K\) satisfies \[ \mathrm{Var}_p|x|^2\le C\,\lambda_p^2\,E_p|x|^2. \] Theorem 1.2 gives a random-projection extension: there are absolute constants \(c_1,c_2,C\) such that if \[ 1\le k\le n^{2/3}/(\log n)^{1/3}, \] then for Haar-random \(E\in G_{n,n-k}\), with probability at least \[ 1-c_1\exp\bigl(-c_2n^{2/3}/(\log n)^{1/3}\bigr), \] the same variance bound holds [1703.09973]. The proof strategy is combinatorial and geometric. Every projection \(K=P_EB_\infty^n\) is partitioned, up to sets of measure zero, into the orthogonal images of the \(2^k\) faces of \(B_\infty^n\) of dimension \(n-k\). A face-by-face decomposition gives \[ E_p f(x)=\sum_F w_F\,E_{p_F}f(x), \] and a two-term expansion yields \[ \mathrm{Var}_p|x|^2 \le \sum_F w_F\,\mathrm{Var}_{p_F}|x|^2 + \sum_F w_F\bigl(E_{p_F}|x|^2-E_p|x|^2\bigr)^2. \] The first term is controlled by reducing each face to an affine image of the \((n-k)\)-cube and invoking known KLS/variance-conjecture bounds for unconditional bodies. The second is controlled by showing that each \(E_{p_F}|x|^2\) differs from the global mean by at most \(O(k)\), and for larger codimension in the random case by applying Gromov–Milman concentration on the Grassmannian together with a union bound over the \(2^k\) faces. The broader significance is stated directly in the paper: prior to this work, the variance conjecture was known for unconditional bodies, the full cube, and for hyperplane \((k=1)\) projections of the cube and cross-polytope. The new results extend those statements to all projections of codimension up to \(\sqrt n\), and with high probability to codimension \(\sim n^{2/3}/(\log n)^{1/3}\). The paper describes them as a step toward verifying the Finite-Variance Extremality Conjecture in ever-larger families of convex bodies [1703.09973]. ## 4. Extremality of translation-invariant phases in the three-state SOS model A different usage of the term arises in the study of Gibbs measures on trees. The model is the SOS model with spin values \(\Phi=\{0,1,2\}\) on the Cayley tree of order two \(\Gamma^2=(V,L)\), where each vertex has three neighbors. A configuration \(\sigma\) assigns a spin \(\sigma_x\in\Phi\) to each \(x\in V\), and the Hamiltonian is \[ H(\sigma)=-J\sum_{\langle x,y\rangle\in L}|\sigma_x-\sigma_y|. \] With \(\beta=1/T\) and \(\theta=e^{\beta J}\), the conventions are: ferromagnetic coupling \(J<0\Rightarrow \theta<1\), antiferromagnetic coupling \(J>0\Rightarrow \theta>1\). Translation-invariant splitting Gibbs measures (TISGMs) are characterized by a constant solution \(z=(z_0,z_1)\) of the fixed-point equations \[ z_i= \Bigl(\sum_{j=0}^{2}\theta^{|i-j|}z_j\Bigr)^2 \Big/ \Bigl(\sum_{j=0}^{2}\theta^{|2-j|}z_j\Bigr)^2, \qquad i=0,1. \] Setting \(x=\sqrt{z_0}\) and \(y=\sqrt{z_1}\) gives the two-dimensional system \[ x=\frac{x^2+\theta y^2+\theta^2}{\theta^2x^2+\theta y^2+1}, \qquad y=\frac{\theta x^2+y^2+\theta}{\theta^2x^2+\theta y^2+1}. \] A proposition of Külske–Rozikov states that for \(\theta>1\) there is a unique positive solution \((x_1,y_1)\). In the ferromagnetic regime \(\theta<1\), the number of solutions jumps according to the critical values \[ \theta_c\approx 0.1414,\qquad \theta_c'\approx 0.2956. \] For \(\theta>\theta_c'\) there is exactly one solution \(v_1=(1,y_1)\); at \(\theta=\theta_c'\), exactly three solutions \(v_1,v_4,v_6\); for \(\theta_c<\theta<\theta_c'\), five solutions \(v_1,v_4,v_5,v_6,v_7\); at \(\theta=\theta_c\), six solutions; and for \(\theta<\theta_c\), seven distinct solutions \(v_i=(x_i,y_i)\), \(i=1,\dots,7\). Each \(v_i\) corresponds to a TI splitting Gibbs measure \(\mu_i\) [1411.5886]. The extremality problem is formulated through reconstruction on the tree. A TISGM \(\mu\) is non-extremal if and only if there is reconstruction of the root spin from the boundary, equivalently if the corresponding tree-indexed Markov chain permits a non-vanishing influence at large depth. Two criteria are used. The Kesten–Stigum spectral condition gives a sufficient condition for non-extremality: \[ k\lambda_2^2>1, \] where \(\lambda_2\) is the second-largest eigenvalue in absolute value of the \(3\times 3\) transition matrix \(P\). The Martinelli–Sinclair–Weitz variance-based condition gives a sufficient condition for extremality: \[ k\kappa\gamma<1, \] where \(\kappa\) is the maximal total-variation contraction per edge and \(\gamma\) bounds the back-influence of the boundary. In this model, \(\kappa=\kappa(x,y)\) can be written explicitly in terms of \(x,y,\theta\), and \[ \gamma(\theta)\le \frac{|1-\theta^2|}{1+\theta^2} \] uniformly over all solutions. The resulting phase diagram is explicit. The phase \(\mu_1\) is extremal if \(\theta<\hat\theta\approx 2.655\) and non-extremal if \(\theta>\check\theta\approx 2.8765\). The phases \(\mu_2,\mu_3\) are always non-extremal whenever they exist. The phases \(\mu_4,\mu_7\) are always extremal in their region of existence. The phases \(\mu_5,\mu_6\) are non-extremal for \(\theta<\theta_*\approx 0.17172\) and extremal for \(\theta>\theta_{**}\approx 0.26586\). Thus three of the seven ferromagnetic states undergo sharp extremality \(\leftrightarrow\) non-extremality transitions at finite critical \(\theta\)’s, two remain extremal for all \(\theta<1\), and two remain non-extremal [1411.5886]. The paper’s own interpretation is explicitly conjectural: informally, the Finite-Variance Extremality Conjecture asserts that for a wide class of finite-state spin-systems on trees, there is a sharp extremality threshold which can be detected purely by a variance-type bound akin to Martinelli–Sinclair–Weitz, or equivalently by a second-moment condition on the tree recursion. The three-state SOS example is presented as evidence because it exhibits finitely computable thresholds and agreement, up to small gaps, between the spectral criterion and the variance criterion [1411.5886]. ## 5. The double Dixie cup problem and exact finite-\(N\) variance extremality In the double Dixie cup problem, for \(N\ge 2\) and \(m\ge 1\), one samples coupon types with probability vector \(p=(p_1,\ldots,p_N)\), \(p_i>0\), \(\sum p_i=1\). Let \[ T_m(N)\equiv T_{m,p} \] be the number of draws needed to collect at least \(m\) copies of each of the \(N\) types, and let \(u=(1/N,\ldots,1/N)\). The conjecture of Doumas and Papanicolaou states that for every \(m\ge 1\), \(N\ge 2\), and every \(p\), \[ \mathrm{Var}_p[T_m(N)]\ge \mathrm{Var}_u[T_m(N)], \] with equality if and only if \(p=u\). Long proves this conjecture exactly, in finite \(N\), by Poissonization and a radial monotonicity argument. In continuous time, coupon \(j\) arrives at rate \(p_j\), and the completion time \(X_{m,p}\) is the maximum of \(N\) independent Erlang\((m,p_j)\) variables. Writing \[ Q_m(x)=P[\mathrm{Erlang}(m,1)>x]=e^{-x}\sum_{a=0}^{m-1}\frac{x^a}{a!}, \] one obtains \[ P(X_{m,p}\le t)=\prod_{j=1}^N [1-Q_m(p_j t)]. \] A coupling with the discrete process gives \(X_{m,p}=S_{T_{m,p}}\), where \(S_k\) is the sum of \(k\) independent \(\mathrm{Exp}(1)\) variables, and therefore \[ E[X_{m,p}^r]=E[T_{m,p}^{(r)}] \] for every integer \(r\ge 1\). In particular, \[ E[T_{m,p}] = \int_0^\infty \Bigl[1-\prod_{j=1}^N(1-Q_m(p_j t))\Bigr]\,dt, \] \[ E[T_{m,p}(T_{m,p}+1)] = 2\int_0^\infty t\Bigl[1-\prod_{j=1}^N(1-Q_m(p_j t))\Bigr]\,dt, \] and \[ \mathrm{Var}_p[T_m(N)] = E[T(T+1)]-E[T]-E[T]^2. \] The key monotonicity theorem is formulated along rays \(p(\theta)=u+\theta h\), \(0\le \theta\le 1\), with \(h\) summing to zero and \(p(1)=p\neq u\). Writing \[ G_\theta(t)=P(X_{m,p(\theta)}\le t)=\prod_{i=1}^N F(q_i t), \] where \(q_i=p_i(\theta)\) and \(F(y)=1-Q_m(y)\), its density is \[ g_\theta(t)=G_\theta(t)\sum_{i=1}^N q_i\phi(q_i t), \] with reverse hazard \(\phi(y)=F'(y)/F(y)\). The derivative measure \[ w_\theta(t)=-\partial_\theta G_\theta(t) \] satisfies \[ w_\theta(t) = \frac{G_\theta(t)\,t}{\theta} \left[ \frac1N\sum_i \phi(q_i t)-\sum_i q_i\phi(q_i t) \right]. \] The bracket is nonnegative by Chebyshev and strictly positive for \(t>0\) because \(p\neq u\). A general lemma states that if \(w_\theta\ge 0\) and the ratio \[ \frac{w_\theta(t)}{t\,g_\theta(t)} \] is increasing in \(t\), then \[ \frac{d}{d\theta}\bigl[\mathrm{Var}(X_{m,p(\theta)})-E[X_{m,p(\theta)}]\bigr]>0. \] Since \(\mathrm{Var}(T)=\mathrm{Var}(X)-E[X]\), this yields the strict increase of \(\mathrm{Var}_p[T_m(N)]\) away from the uniform vector. The single-site input is Lemma 4.5: for \(F\) the CDF of \(\mathrm{Gamma}(m,1)\) and \(\phi=F'/F\), \[ e(y)=\frac{d}{d\log y}[\log \phi(y)] \] is strictly negative and strictly decreasing on \((0,\infty)\). Equivalently, for any \(c>1\), the ratio \(\phi(cy)/\phi(y)\) is strictly decreasing in \(y\). This implies the monotonicity of \(w_\theta/(t g_\theta)\), and Theorem 4.7 follows: for every nonuniform \(p\), \[ \mathrm{Var}_p[T_m(N)]>\mathrm{Var}_u[T_m(N)]. \] Hence the uniform vector uniquely minimizes the variance [2604.25108]. The same paper also obtains asymptotic results in the equal-probability case \(p_i=1/N\), allowing fixed or growing \(m=m_N\). If \(b_N\) solves \[ NQ_{m_N}(b_N)=1 \] and \[ a_N=\frac{Q_{m_N}(b_N)}{f_{m_N}(b_N)}, \qquad f_m(x)=e^{-x}x^{m-1}/(m-1)!, \] then \[ \frac{T_{m_N}(N)-Nb_N}{Na_N}\Rightarrow G, \] where \(G\) is a standard Gumbel law, and \[ E[T_{m_N}(N)]=Nb_N+\gamma Na_N+o(Na_N), \qquad \mathrm{Var}(T_{m_N}(N))=\frac{\pi^2}{6}N^2a_N^2+o(N^2a_N^2). \] For fixed \(m\), \[ b_N=\log N+(m-1)\log\log N-\log((m-1)!)+o(1), \qquad a_N\to 1, \] so \[ E[T_m(N)] = N\log N+(m-1)N\log\log N+N[\gamma-\log((m-1)!)] + o(N), \] and \[ \mathrm{Var}(T_m(N))\sim (\pi^2/6)N^2. \] The terminal-defect transfer theorem further extends the analysis to unequal probabilities, including power-law choices \(p_{N,j}\propto j^{-\alpha}\), where an endpoint-Laplace analysis again yields a Gumbel limit [2604.25108]. ## 6. Comparative interpretation, methodological patterns, and open status A common source of confusion is terminological. The corpus suggests that *Finite-Variance Extremality Conjecture* functions less as a single conjecture than as a recurring extremal paradigm. In one formulation it posits a universal Gaussian lower bound for central probability mass at one standard deviation; in another it is the variance conjecture for log-concave measures; in another it concerns reconstruction thresholds for Gibbs phases on trees; and in another it is a finite-\(N\) minimization theorem for coupon-collector variance. Despite these differences, the methodological patterns are strikingly parallel. The Gamma argument reduces by scaling to a single-parameter family, proves monotonicity by integral comparison, and identifies the infimum through a central-limit limit. The cube-projection argument decomposes a high-dimensional object into lower-dimensional faces, controls variance by a two-term expansion, and then uses concentration on the Grassmannian. The SOS analysis translates phase extremality into a tree-indexed Markov-chain problem and combines a spectral criterion with the Martinelli–Sinclair–Weitz variance-based bound. The double Dixie cup proof Poissonizes the process, rewrites the completion time as a maximum of Erlang variables, and establishes radial variance monotonicity through a size-biased monotone-likelihood-ratio argument based on reverse-hazard monotonicity. The status of the various formulations is mixed. In the Gamma setting, the universal finite-variance statement remains conjectural, but the Gamma-family theorems provide an exact model case and identify the Gaussian limit as the extremizing limit. In convex geometry, the variance conjecture remains open in full generality, but significant families—such as projections of the cube in the regimes stated above—are now covered. In the tree-spin setting, the three-state SOS model provides a detailed worked example in which variance-based methods locate all extremality transitions up to small gaps. In the double Dixie cup problem, by contrast, the finite-variance extremality conjecture of Doumas and Papanicolaou is fully proved: for every \(m\ge 1\), \(N\ge 2\), and every positive coupon-probability vector, the uniform vector is the unique minimizer of \(\mathrm{Var}[T_m(N)]\). A plausible implication is that the phrase is best understood as naming a research program centered on extremality under finite second-moment control. The specific extremizer—Gaussian, isotropic, uniform, or a phase singled out by reconstruction criteria—depends on the ambient structure. What unifies the literature is the claim that finite-variance information is not merely quantitative bookkeeping: it can determine the extremal object itself, or at least sharply constrain the location of extremality. Source: https://www.emergentmind.com/topics/finite-variance-extremality-conjecture