Universality of Persistence of Random Polynomials

In this paper, we consider the random polynomials

where $a_{i}=\sqrt{R(i)}\xi_{i}$, with $\{\xi_{i}\}_{0\leq i\leq n}$ i.i.d. random variables $\mathbb{E}[\xi_{i}]=0$ and $\mathrm{Var}(\xi_{i})=1$. Hhere $R(\cdot):\mathbb{Z}_{+}\mapsto(0,\infty)$ is a regularly varying function of order $\alpha>-1$, with $R(0)=1$. The regularly varying assumption implies we can write $R(i)=i^{\alpha}L(i)$ for some $\alpha\in\mathbb{R}$, and $\{L(i)\}_{i\geq 0}$ is slowly varying at infinity, i.e.,

We seek to study the decay of the persistence probability, i.e., the probability of $Q_{n}$ having no real zero. To this goals, we introduce the following:

Since an odd degree polynomial necessarily has a real root, we have $p_{n}^{(\alpha)}=0$ for $n$ odd. For $n$ even, the quantity $\log p^{(\alpha)}_{n}/\log n$ is anticipated to converge to a constant as $n\to\infty$. This constant is known as the persistence exponent of the random polynomial $Q_{n}$. In the scenario when $R(i)=1$, i.e. $a_{i}$ are i.i.d. random variables,—Poonen and Stoll [PS99, Section 9.1] conjectured that this constant remains invariant across different distributions of $a_{i}$, provided that $a_{i}$ satisfy the following conditions: (1) mean zero, (2) $P(a_{i}\neq 0)>0$, and (3) $a_{i}$ belongs to the domain of attraction of the normal distribution. Essentially, this conjecture suggests that the persistence exponent of $Q_{n}$ is universal within the class of i.i.d. random variables with zero mean and finite variance.

Our main result, outlined below, resolves this conjecture and extends it by demonstrating the universality of persistence exponents for $Q_{n}$ across a significantly broader range of random polynomial classes.

Significant strides have been made in the investigation of persistence in random polynomials, starting with [DPSZ02], where it was shown that when the coefficients $a_{i}$ of $Q_{n}$ are i.i.d. and possess finite moments of all orders, the persistence exponents of $Q_{n}$ are independent of the distribution of $a_{i}$ and are expressed as $-4b_{0}$. This seminal result laid the foundation for subsequent research in this field. In [DM15], the persistence probability was examined under the assumption that the coefficients of random polynomials are independent Gaussian variables with mean zero and variances that vary regularly. It was demonstrated therein that the persistence exponent in this context is given by $-2b_{\alpha}-2b_{0}$, as stated in Theorem 1.1.

The persistence probabilities of random polynomials and other stochastic systems have also been closely scrutinized in the physics literature; see, for instance, [Maj99, SM07, SM08] for further exploration. In a recent contribution [PS18], the authors computed $b_{0}=\frac{3}{16}$ precisely by evaluating the probability of non-real eigenvalues within the random orthogonal matrix ensemble [KPT⁺16]. In fact, [PS18] found also an alternative way to arrive at the same exponent by matching the persistence probabilities between a 2-dimensional heat equation started from random initial data (studied in [DM15]) and the Glauber dynamics of a semi-infinite Ising spin chain (studied in [DHP95, DHP96]).

The exploration of the zero-count of random polynomials has a rich historical backdrop. A particularly well-studied scenario involves the coefficients ${a_{i}}_{i\geq 1}$ being independent and identically distributed as standard Gaussian random variables. Early investigations, dating back to the work of Littlewood and Offord [LO38, LO39, LO43], yielded both upper and lower bounds for the expected total number of real roots, denoted as $N_{n}$. Their contributions also provided insights into the tails of this distribution, offering an initial estimate that $\mathbb{P}(N_{n}=0)$ scales as $O\big{(}\frac{1}{\log n}\big{)}$ when the coefficients $a_{i}$ follow the distribution $\mathbb{P}(a_{i}=-\frac{1}{3})=\mathbb{P}(a_{i}=0)=\mathbb{P}(a_{i}=1)=\frac{1}{3}$.

Kac [Kac43] made a significant stride by deriving the exact formula for the expected value of $N_{n}$ when $a_{i}$’s are independent and identically distributed as standard normal random variables. Subsequently, over several decades, researchers honed in on the precise asymptotic behavior of $\mathbb{E}[N_{n}]$, refining and extending Kac’s findings. In a more recent development, as highlighted by [NNV16], it was demonstrated that the asymptotic expectation of $\mathbb{E}[N_{n}]$ equals $\frac{2}{\pi}\log n+O(1)$ for the case where $a_{i}$ follows a distribution with $\mathbb{E}[a_{i}]=0$ and $\mathbb{E}[a_{i}^{2+\varepsilon}]<\infty$.

The challenge of determining the probability of a random polynomial having no real roots remained a formidable task until the seminal work of [DPSZ02]. Their work established that for even values of $n$, $\mathbb{P}(N_{n}=0)$ exhibits polynomial decay, specifically as $n^{-4b_{0}+o(1)}$. They also showed that the probability of $N_{n+k}$ to have exact $k$ (simple) zeros also behave in the same way. The study of non-identically distributed, but independent coefficients has been considered in [DM15], where the authors verify Theorem 1.1, under the assumption that all coefficients are Gaussian. Theorem 1.1 represents a significant generalization of the findings in [DM15] and [DPSZ02]. It achieves this by removing the requirement that $a_{i}$ possess finite moments of all orders or follow Gaussian distributions. In other word, it shows universality of the persistence of random polynomials. This pivotal development resolves Poonen and Stole’s conjecture on persistence of random polynomials.

The study of random polynomials has been a dynamic field, continually yielding fresh insights and relevant methodologies. A significant portion of this research has been devoted to investigating the presence of real roots in random polynomials. In particular, the problem of a random polynomial having no real zeros has attracted widespread attention within the mathematics and physics communities.

Many prior approaches to establish the persistence of such random polynomials have leaned heavily on comparisons with Gaussian distributions, often employing techniques like Komlos-Major-Tusnady embedding and tools such as Slepian’s inequality [DPSZ02, DM15]. However, a notable challenge in establishing the universality of this persistence lies in the absence of KMT-type results for random variables with only finite second moments. Moreover, the challenge becomes even more complex when dealing with non-identically distributed random variables. We seek to resolve this by innovating a new approach for showing the persistence. We divide the proofs into two parts which are given as follows:

We show $\mathfrak{LimSup}$ and $\mathfrak{LimInf}$ in Sections 3 and 4 respectively. The proof ideas for showing these two claims are different and in what follows, we discuss those proof ideas in details. We divide the proofs of $\mathfrak{LimSup}$ and $\mathfrak{LimInf}$ into two steps. We begin with the steps of the proof of $\mathfrak{LimSup}$ and then proceed to the steps in the proof of $\mathfrak{LimInf}$. From this point forward, we assume that $n$ is an even integer. However, it turns out that the proofs concerning $\mathfrak{LimSup}$ and $\mathfrak{LimInf}$ do not make use of this assumption, with the exception of Propositions 4.5 and 4.4.

Now, we need to analyze the probability of no real roots of $Q_{n}$ in each of these four intervals. However, it is important to note that the events of having no zeros in these intervals are not independent. To address this challenge, we demonstrate that the probability of having no real zeros of $Q_{n}$ in disjoint intervals $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3},\mathfrak{D}_{4}$ are dictated by the disjoint components of $Q_{n}$. For instance, the probability of no zero of $Q_{n}$ in the interval $\mathfrak{D}_{1}$ is mainly governed by $\sum_{i\in[n^{\delta},n^{1-\delta}]}a_{i}x^{i}$ and this holds roughly because the variance of $\sum_{i\in[n^{\delta},n^{1-\delta}]}a_{i}x^{i}$ is greater than any other portions of $Q_{n}$ when $x\in\mathfrak{D}_{1}$ (see Theorem ). Consequently, we can express the probability of having no real zeros in any of these four intervals as approximately the product of the probabilities of having no real zeros in each of these intervals.

We now explain why the probability of no zeros for the intervals $\mathfrak{D}_{1}$ and $\mathfrak{D}_{2}$ are different and how we bound those probabilities. Recall that the probability of no real zero of $Q_{n}$ in the interval $\mathfrak{D}_{1}$ depends on $\sum_{i\in[n^{\delta},n^{1-\delta}]\cap\mathbb{Z}}a_{i}x^{i}$ whereas the probability of no real zero of $Q_{n}$ in $\mathfrak{D}_{2}$ is governed by $\sum_{i\in[n-n^{1-\delta},n-n^{\delta}]\cap\mathbb{Z}}a_{i}x^{i}$. We further divide the polynomial $\sum_{i\in[n^{\delta},n^{1-\delta}]\cap\mathbb{Z}}a_{i}x^{i}$ into almost $\log n$ number of sub-polynomials $\sum_{i\in\mathcal{I}^{(-)}_{r}}a_{i}x^{i}$ (see (3.1) for $\mathcal{I}^{(-)}_{r}$) of varying sizes. Each of these sub-polynomials (for instance, $\sum_{i\in\mathcal{I}^{(-)}_{r}}a_{i}x^{i}$) when restricted to a particular sub-interval ($\mathcal{J}^{(-)}_{r}$ of (3.3) in the case of $\mathcal{I}^{(-)}_{r}$) of $\mathfrak{D}_{1}$, converges to a Gaussian process which is directly linked to $\{Y^{(\alpha)}_{t}\}_{t\geq 0}$. These sub-polynomials are constructed in Section 3.1 (see (3.4)). Such construction also ensures that those sub-polynomials when restricted to some other sub-intervals of $\mathfrak{D}_{1}$, the covariance function decays (see Lemma 3.5). As a result, the probability of no real zeros of $\sum_{i\in[n^{\delta},n^{1-\delta}]\cap\mathbb{Z}}a_{i}x^{i}$ in any of those sub-intervals of $\mathfrak{D}_{1}$ can be bounded by the probability of the respective sub-polynomials not exceeding an arbitrary small negative number (Lemma 3.2). Since all these sub-polynomials are independent of each other and they weakly converges (when restricted to the respective sub-intervals) to the same Gaussian process, we do have

where $K=O(\log n/\log M)$ for any fixed $M>0$. The constant of proportionality in the above equation is $o(\log n)$. Taking logarithm in both sides in the above display, dividing by $\log n$ and letting $n\to\infty$ shows that probability of no real zeros in $\mathfrak{D}_{1}$ is bounded by $n^{-b_{\alpha}+o(1)}$. On the other hand, to bound the similar probability for $\mathfrak{D}_{2}$, we divide $\sum_{i\in[n-n^{1-\delta},n-n^{\delta}]\cap\mathbb{Z}}a_{i}x^{i}$ into almost $\log n$ many sub-polynomials corresponding to index sets $\mathcal{I}^{(+)}_{r}$ (see (3.2)). However, unlike the $\mathfrak{D}_{1}$ case, each of these sub-polynomials (for instance, $\sum_{i\in\mathcal{I}^{(+)}_{r}}a_{i}x^{i}$) when restricted to a particular sub-interval ($\mathcal{J}^{(+)}_{r}$ of (3.3) in the case of $\mathcal{I}^{(+)}_{r}$) of $\mathfrak{D}_{2}$, converges to a Gaussian process which is directly linked to $\{Y^{(0)}_{t}\}_{t\geq 0}$. Now, by a similar argument, we show

where the right hand side of the above display decays as $n^{-b_{0}+o(1)}$. Combining these probabilities for $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$ and $\mathfrak{D}_{4}$ yields the proof of $\mathfrak{LimSup}$. Although the steps to find the exact asymptotics are straightforward, the assumption of only finite second moments enforces careful accounting of probability bounds which is executed mainly in the proof of Lemma 3.2 through employing a combinatorial selection criterion ‘move, flush and repeat’ and the required probability bounds are assimilated from several important lemmas stated and proved in Section 3.2.2.

Like as in the proof of $\mathfrak{LimSup}$, we notice that $\mathds{P}(Q_{n}(\pm e^{u})<0,u\in A_{r})$ is dictated by the sub-polynomials $Q^{(r)}_{n}(x):=\sum_{i\in B_{r}}a_{i}x^{i}$ (see (4.2) for the definitions of the sets $B_{r}$). From the definition of $Q^{(r)}_{n}(\cdot)$, it follows that $Q_{n}(x)=\sum_{r\in\mathcal{R}}Q^{(r)}_{n}(x)$. Such decomposition of $Q_{n}(x)$ implies

where $\sigma_{n}(u)$ is the standard deviation of the random variable $Q_{n}(e^{u})$. Since the sub-polynomials $Q^{(r)}_{n}$ are independent, the probability of the event of the right hand side is equal to the product of the the following probabilities;

The main thrust of the proof of $\mathfrak{LimInf}$ lies in the lower bound of the above probabilities. Indeed, we derive these lower bounds in Proposition 4.1, 4.2, 4.3, 4.5 and 4.4 respectively. We show in Proposition 4.1 and  4.2, that the above probabilities for $r=-1$ and $r=1$ are bounded below by the probabilities of no-zero crossing of two Gaussian processes which later is bounded by the non-zero crossing probabilities of the Gaussian processes $Y^{(\alpha)}_{t}$ and $Y^{(0)}_{t}$ respectively (through Lemma 4.22). Thus the probabilities of the above display for $r=-1$ and $r=1$ roughly contributes $n^{-2b_{\alpha}}$ and $n^{-2b_{0}}$ in the lower bound to the probability of $Q_{n}(x)<0$ for all $x\in\mathbb{R}$. Furthermore, Propositions 4.3, 4.5 and 4.4 shows that the probabilities of the above display corresponding to $r=0,-2,2$ are bound below by $n^{-o(1)}$. Assembling these bounds together yields the proof of $\mathfrak{LimInf}$. In the next step, we discuss the ideas that we use in lower bounding the probabilities in the above display for $r=-1$ and $r=1$. It is important to note that, while the probabilities in (2.3) for $r=-2,2$ do not directly contribute to the exact value of the persistence exponent, the proof of their lower bound (which is of the order $n^{-o(1)}$) in Propositions 4.5 and 4.4 relies heavily on the fact that $Q_{n}$ is a polynomial of even degree.

We discuss here strategies to bound (2.3) for $r=-1,1$. As in the proof of $\mathfrak{LimSup}$, we divide the intervals $A_{-1}$ and $A_{+1}$ into $T_{n}=O(\log n)$ many intervals. These are now denoted as $\{\tilde{\mathcal{J}}^{(-)}_{p}:1\leq p\leq T_{n}\}$ and $\{\tilde{\mathcal{J}}^{(+)}_{p}:1\leq p\leq T_{n}\}$ respectively. On the other hand, the index sets $B_{-1}$ and $B_{+1}$ are also divided into $\{\tilde{\mathcal{I}}^{(-)}_{p}:1\leq r\leq T_{n}\}$ and $\{\tilde{\mathcal{I}}^{(+)}_{p}:1\leq r\leq T_{n}\}$ (see (4.3) and (4.4)). We show that the probability in (2.3) could be lower bounded by product of certain probabilities of the polynomials $\tilde{Q}^{(-1),p}_{n}(x)=\sum_{i\in\tilde{\mathcal{I}}^{(-)}_{p}}a_{i}x^{i}$ when $r=-1$ and $\tilde{Q}^{(-1),p}_{n}(x)=\sum_{i\in\tilde{\mathcal{I}}^{(-)}_{p}}a_{i}x^{i}$ when $r=+1$. We show below how this bound follows for $r=-1$ case:

where $B_{1\cdot},B_{2\cdot},B_{3\cdot}$ and $B_{4\cdot}$ are defined in (4.40), (4.41), (4.42) and (4.43) respectively. In words, $B_{1p}$ signifies the event that $\tilde{Q}^{(-1),p}_{n}(e^{u})$ (when scaled by $\sigma_{n}(u)$) to be less then a small negative constant for all $u$ inside and around a neighborhood of $\mathcal{J}^{(-)}_{p}$ whereas $B_{2\cdot},B_{3\cdot}$ and $B_{4\cdot}$ signifies $\tilde{Q}^{(-1),p}_{n}(e^{u})/\sigma_{n}(e^{u})$ to be less than positive numbers which decays as $u$ moves away $\mathcal{J}^{(-)}_{p}$. We show that $\mathds{P}(B_{1p}\cap B_{2p}\cap B_{3p}\cap B_{4p})\approx\mathds{P}(B_{1p})$ and $\mathds{P}(B_{1p})$ can be lower bounded by probability of the Gaussian process $Y^{(\alpha)}_{t}$ being smaller a small negative number everywhere inside an interval whose length grows as $ne^{-T_{n}}$. This last probability is indeed same as the no-zeros crossing probability of the Gaussian process $Y^{(\alpha)}_{t}$ when the small constant is taken to $0$. We show the above mentioned lower bound by combining Lemma 4.8, Lemma 4.9, Lemma 4.10 with Lemma 4.22. The first three lemma required careful derivation of the decay of probabilities of $\tilde{Q}^{(-1),p}_{n}(e^{u})/\sigma_{n}(e^{u})$ exceeding a positive constants as $u$ moves away from $\mathcal{J}^{(-)}_{p}$. Since we have assumed only second moments of $a_{i}$’s finite, the derivation of these probabilities poses several challenges which are overcome in Lemma 4.6. Taking the product of the aforementioned lower bound over all $p\in[1,T_{n}]\cap\mathbb{Z}$ shows that the probability in (2.3) for $r=-1$ can be bounded from below by the product of $T_{n}$ many no-zero crossing probability of $Y^{(\alpha)}_{t}$. Similarly, the probability in (2.3) for $r=+1$ can be bounded from below by the product of $T_{n}$ many no-zero crossing probability of $Y^{(0)}_{t}$. These explain the bounds obtained in Proposition 4.1 and 4.2 which are the most import components in the proof of $\mathfrak{LimInf}$.