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1 Introduction↩︎

1.1 Random Data Theory and Rogue Waves as Extreme Events↩︎

The probabilistic study of dispersive partial differential equations originates from the seminal works of Lebowitz–Rose–Speer and Bourgain on invariant measures for the nonlinear Schrödinger equation; see [2]–[4]. These developments initiated a systematic investigation of low-regularity dynamics through randomization of either forcing terms or initial data.

From a structural point of view, such problems naturally split into two classes. The first concerns stochastic partial differential equations (SPDEs), where randomness enters through singular forcing and leads to dynamics such as the KPZ equation [5] and stochastic quantization models in Euclidean quantum field theory [6]. The second class consists of deterministic dispersive equations with random initial data, which provide a probabilistic framework for studying low-regularity behavior and are closely connected to wave turbulence theory [7], [8]. In many singular settings, a common feature is that the underlying analysis requires renormalization to handle divergent nonlinear interactions.

In the past decade, there has been remarkable progress in the analysis of singular stochastic parabolic equations. The development of the theory of regularity structures by Hairer [9], [10] and paracontrolled calculus by Gubinelli–Imkeller–Perkowski [6] has led to the local well-posedness theory, essentially completing the picture in the so-called subcritical regime; see [11] for further introduction.

More recently, the emerging theories of random tensors [11] and random averaging operators [12], developed by Deng–Nahmod–Yue, can be interpreted as dispersive counterparts to the aforementioned parabolic frameworks. These theories provide a systematic frameworks to the renormalization of oscillatory multilinear interactions; see also the corresponding hyperbolic counterpart [13].

Random data theory has played a fundamental role in the study of the long-time behaviour of solutions to evolution partial differential equations. One of the most active directions in this field is the derivation of the Wave Kinetic Equation (WKE), which connects the statistical behaviour of Fourier coefficients of solutions to an effective kinetic description. Significant recent advances in this direction can be found in [14]–[18], together with the references therein.

Another interesting direction related to random data theory concerns the so-called rogue waves, a term used by oceanographers to describe isolated and large-amplitude waves, typically defined as those whose height exceeds twice the significant wave height of the ambient sea state. Empirically, such events occur more frequently than predicted by Gaussian statistics; see [1], [19], [20].

To explain this phenomenon, several mechanisms for rogue wave formation have been proposed, although no unified theory is currently available. Linear superposition and nonlinear focusing are two fundamental mechanisms widely studied in the literature. The former is a constructive interference phenomenon: the phases of many weakly interacting waves align at a certain point in space, producing a large-amplitude peak. The latter suggests that rogue waves may emerge from a significant exchange of energy between waves of different wavenumbers, leading to a substantial amplification of one component; see [21]. In this setting, rogue waves are often viewed as limiting cases of breather solutions and may also arise from soliton collisions; see [19]. These represent non-generic behaviours and may be regarded as rare phenomena, or more generally as extreme events from a probabilistic perspective; see [1].

The probabilistic perspective suggests that rogue waves provide a canonical, yet still poorly understood, example of extreme events. A central problem is therefore to quantify deviations of the wave-height distribution from Gaussianity, and thereby to estimate the likelihood of rogue wave occurrence. This remains a long-standing problem with significant implications for ships and naval structures; see [20].

In [20], Dematteis, Grafke and Vanden-Eijnden observed that rogue waves obey a large deviations principle. This phenomenon was later rigorously established by Garrido et al. for the cubic weakly nonlinear Schrödinger (NLS) equation \[\begin{align} \label{eq:snls} \mathrm{i}\partial_t u + \partial_{xx} u + {\varepsilon^{\alpha}} |u|^{2} u = 0,\quad \alpha>1, \end{align}\tag{1}\] with randomized periodic initial data of the form \[u(0,x) = \sum_{n \in \mathbb{Z}} c(n)\, g_{n}\, e^{\mathrm{i} n x}.\] Their result holds over time scales \(t \sim \varepsilon^{-\beta}\) for both subcritical and critical cases, that is, \(\alpha - 1 > \beta\) and \(\alpha - 1 = \beta > 0\) respectively. Their analysis relies on an exponential decay assumption on the deterministic Fourier components of the random initial data, namely \(c(n)=a e^{-b|n|}\) or \(c(n)=a e^{-b|n|^2}\) for all \(n \in \mathbb{Z}\), where \(a,b>0\) are fixed. This assumption plays a crucial role in their argument; see [1] for further details. Recently, this condition has been relaxed. In [22], Liang and Wang extended the result to \(\ell^1(\mathbb{Z})\) decay, while Fan and Ye [23] further improved it to \(c_n = \langle n \rangle^{-\frac{1}{2}+}\).

In contrast to standard NLS 1 , Grande studied the following beating NLS \[\begin{align} {\mathrm{i}}\partial_t u + \partial_{xx} u = 2\cos(2x)\,|u|^2 u, \end{align}\] with randomized periodic initial data supported on two Fourier modes of the form \[u(0,x) = \varepsilon \bigl(\alpha e^{\mathrm{i}x} + \beta e^{-\mathrm{i}x}\bigr),\] where \(\alpha\) and \(\beta\) are complex-valued independent Gaussian random variables with zero mean and variances \(\sigma_\alpha^2 = \mathbb{E}\bigl[|\alpha|^2\bigr]\) and \(\sigma_\beta^2 = \mathbb{E}\bigl[|\beta|^2\bigr]\). When \(\sigma^2_\alpha \neq \sigma^2_\beta\), he showed that resonant energy exchange between Fourier modes leads to a fattening of the tails of the probability distribution of the sup-norm of the solution, thereby increasing the likelihood of rogue wave formation. Consequently, a large deviations principle holds; see [21].

The above discussion has been confined to the spatially periodic setting in the study of rogue waves. As noted by Wilkening and Zhao [24], rogue waves are also intimately connected with spatially quasi-periodic dynamics arising from modulational instability (also known as Benjamin–Feir instability). Thus, extending the study of rogue waves to the spatially quasi-periodic setting under randomization remains a natural and largely unexplored direction. In addition, the deterministic Cauchy problem with spatially quasi-periodic initial data forms an important and active area in nonlinear partial differential equations; see, for example, the works of Deift [25], [26] and subsequent works citing them.

1.2 Randomized Quasi-Periodic Fourier Series on \(\mathbb{R}^d\) and Initial Data↩︎

Before formulating the problem precisely, we first introduce in this subsection the randomization of spatially quasi-periodic Fourier series on \(\mathbb{R}^d\).

A function \(f:\mathbb{R}^d \to \mathbb{C}\) is said to be quasi-periodic if it is quasi-periodic in each spatial direction \(x_j\), with associated rationally independent frequency vector \(\omega_j \in \mathbb{R}^{\nu_j}\) for \(j=1,\dots,d\), that is, \[\begin{align} \label{eq:rationallyindependent} \langle n_j,\omega_j \rangle= 0 \quad \Longrightarrow \quad n_j = 0 \in \mathbb{Z}^{\nu_j}. \end{align}\tag{2}\]

Let \(\nu := \sum_{j=1}^d \nu_j\) denote the total number of frequencies. We define the frequency matrix \(\boldsymbol{\Omega} \in \mathbb{R}^{\nu \times d}\) (with \(\nu > d\)) by \[\begin{align} \label{eq:frequencymatrix} \boldsymbol{\Omega} = \mathrm{diag}\bigl(\omega_1^{\top}, \dots, \omega_d^{\top}\bigr) = \begin{pmatrix} \omega_1^{\top} & & \\ & \ddots & \\ & & \omega_d^{\top} \end{pmatrix}. \end{align}\tag{3}\] We say that \(\boldsymbol{\Omega}\) is non-resonant or rationally independent if 2 holds for each direction. Equivalently, in a more compact form, \[\boldsymbol{n}\boldsymbol{\Omega} = \boldsymbol{0} \quad \Longrightarrow \quad \boldsymbol{n} = \boldsymbol{0} \in \mathbb{Z}^{\nu},\] where \(\boldsymbol{n} = (n_{1},\dots,n_d) \in \mathbb{Z}^{\nu_1} \times \cdots \times \mathbb{Z}^{\nu_d} \simeq \mathbb{Z}^{\nu}\).

The block-diagonal structure of \(\boldsymbol{\Omega}\) reflects the independence of the frequency components across different spatial directions. In particular, initial data [eq:initialdata] is quasi-periodic in each spatial direction \(x_j\) with frequency vector \(\omega_j\). Therefore, \(f\) admits the quasi-periodic Fourier expansion \[\label{eq:fourierseries} f(\boldsymbol{x}) = \sum_{\boldsymbol{n} \in \mathbb{Z}^\nu} \hat{f}(\boldsymbol{n}) \, e^{\mathrm{i}\langle \boldsymbol{n}\boldsymbol{\Omega}, \boldsymbol{x} \rangle}, \quad \boldsymbol{x} \in \mathbb{R}^d,\tag{4}\] where the Fourier coefficients \(\hat{f}(\boldsymbol{n})\) are determined by the spatial average \[\label{eq:fouriercoefficient} \hat{f}(\boldsymbol{n}) = \lim_{L \to \infty} \frac{1}{(2L)^d} \int_{[-L,L]^d} f(\boldsymbol{x}) \, e^{-\mathrm{i}\langle \boldsymbol{n}\boldsymbol{\Omega}, \boldsymbol{x} \rangle} \, \mathrm{d}\boldsymbol{x}.\tag{5}\]

By introducing the angular variables \(\boldsymbol{y} = \boldsymbol{x}\boldsymbol{\Omega}^\top \pmod{2\pi}\) on the torus \(\mathbb{T}^\nu\), we define the generating function \(F\) as the following Fourier expansion on \[F(\boldsymbol{y}) = \sum_{\boldsymbol{n} \in \mathbb{Z}^\nu} \hat{f}(\boldsymbol{n}) e^{\mathrm{i} \langle \boldsymbol{n}, \boldsymbol{y}\rangle}, \quad \boldsymbol{y} \in \mathbb{T}^\nu,\] such that \(f(\boldsymbol{x}) = F(\boldsymbol{x}\boldsymbol{\Omega}^\top)\).

By the Birkhoff Ergodic Theorem, the rational independence of frequency matrix \(\boldsymbol{\Omega}\) ensures that spatial average in 5 coincides with the Haar integral over the torus, that is, \[\hat{f}(\boldsymbol{n}) = \frac{1}{(2\pi)^\nu} \int_{\mathbb{T}^\nu} F(\boldsymbol{y}) e^{-\mathrm{i} \langle \boldsymbol{n}, \boldsymbol{y} \rangle} \,\mathrm{d}\boldsymbol{y}.\]

In this paper, we consider the randomization of Fourier series 4 by prescribing the coefficients as \[\hat{f}(\boldsymbol{n}) = c(\boldsymbol{n}) g_{\boldsymbol{n}}, \quad \boldsymbol{n} \in \mathbb{Z}^\nu.\]

• Here \(\{c(\boldsymbol{n})\}_{\boldsymbol{n}\in\mathbb{Z}^\nu}\) is a deterministic sequence satisfying the following polynomial decay condition: \[\begin{align} \label{eq:decay} |c(\boldsymbol{n})| \le \tnorm{\boldsymbol{n}}_{-\rho} = \prod_{j=1}^{d}\prod_{j'=1}^{\nu_j} (1+|n_{j,j'}|)^{-\rho_{j,j'}}, \end{align}\tag{6}\] where \[\boldsymbol{n} = (n_{1,1},\cdots,n_{1,\nu_1};\cdots;n_{d,1},\cdots,n_{d,\nu_d}) \in \mathbb{Z}^{\nu_1} \times \cdots \times \mathbb{Z}^{\nu_d},\] and \[\rho = (\rho_{1,1},\cdots,\rho_{1,\nu_1};\cdots;\rho_{d,1},\cdots,\rho_{d,\nu_d}) \quad \text{with } \rho_{j,j'} > 0 ~~\text{large enough for all indices}.\]

• The random component \(\{g_{\boldsymbol{n}}\}_{\boldsymbol{n} \in \mathbb{Z}^\nu}\) constitutes a family of independent and identically distributed (i.i.d. for short) standard complex Gaussian random variables satisfying the following conditions: \[\mathbb{E}[g_{\boldsymbol{n}}] = 0, \qquad \mathbb{E}[g_{\boldsymbol{n}}g_{\boldsymbol{m}}] = 0, \qquad\mathbb{E}[g_{\boldsymbol{n}} \overline{g_{\boldsymbol{m}}}] = \delta_{\boldsymbol{n},\boldsymbol{m}},\] where \(\delta_{\boldsymbol{n},\boldsymbol{m}}\) denotes the Kronecker delta. Equivalently, \(\operatorname{Re}\, g_{\boldsymbol{n}}\) and \(\operatorname{Im}\, g_{\boldsymbol{n}}\) are independent \(\mathcal{N}_{\mathbb{R}}(0,\tfrac{1}{2})\) random variables; see 5.5.1 for further details.

This construction defines a Gaussian random field \(F\) on the torus \(\mathbb{T}^\nu\), which induces a randomized quasi-periodic field \(f\) on \(\mathbb{R}^d\), that is, \[\label{eq:randomqp95d} f(\boldsymbol{x}) = \sum_{\boldsymbol{n} \in \mathbb{Z}^\nu} c(\boldsymbol{n}) g_{\boldsymbol{n}} e^{\mathrm{i} \langle \boldsymbol{n}\boldsymbol{\Omega}, \boldsymbol{x} \rangle}.\tag{7}\]

1.3 Problem Statement and Main Results↩︎

Motivated by the preceding discussion, we consider the Cauchy problem on \(\mathbb{R}^d\) for a weakly nonlinear Schrödinger equation (NLS) with randomized quasi-periodic initial data:

Here \(u = u(t,\boldsymbol{x})\) is a complex-valued function with \((t,\boldsymbol{x}) \in \mathbb{R} \times \mathbb{R}^d\). The operator \(\partial_t\) denotes the time derivative, and \(\Delta = \sum_{j=1}^d \partial_j^2\) is the Laplacian, where \(\partial_j = \frac{\partial}{\partial x_j}\). The parameter \(0 < \varepsilon \ll 1\) measures the strength of the nonlinearity, and \(\mathrm{i}\) denotes the imaginary unit satisfying \(\mathrm{i}^2 = -1\).

For model [eq:nls]-[eq:initialdata], we study the corresponding rogue waves introduced above. Although evolution equation [eq:nls] is deterministic, randomness in initial data [eq:initialdata] induces a stochastic process \(t \mapsto u(t,\boldsymbol{x})\) for each fixed \(\boldsymbol{x} \in \mathbb{R}^d\), and a random field \(\boldsymbol{x} \mapsto u(t,\boldsymbol{x})\) for each fixed \(t \in \mathbb{R}\).

We quantify such extreme events by introducing the tail event \[\label{eq:rogue} \mathscr{A}_\varepsilon^\infty = \left\{ \sup_{\boldsymbol{x} \in \mathbb{R}^d} |u(t, \boldsymbol{x})| > z_0 {\varepsilon^{-\frac{1}{2}}} \right\},\tag{13}\] where \(z_0 > 0\) is a fixed threshold. This event describes the occurrence of a wave of amplitude at least \(z_0 \varepsilon^{-1/2}\) at a fixed time \(t>0\).

The main objective of this paper is to describe the asymptotic behavior of the probability \(\mathbb{P}(\mathscr{A}_\varepsilon^\infty)\) as \(\varepsilon \to 0^+\).

Our main result is a nonlinear large deviations principle in the subcritical time regime (1), together with a linear large deviations principle (2) and a well-posedness result (3).

1.4 Outline of the paper↩︎

In 2, we establish a linear LDP for the associated linear problem. In 3, we prove a well-posedness result, including the relevant time scale and Fourier decay properties. In 4, we derive the Duhamel estimates and establish the nonlinear LDP. In 5, we collect some basic facts on probability theory related to complex Gaussian random variables.

1.5 Notation↩︎

We summarize some notation used throughout the paper below.

2 Linear LDP↩︎

Consider the linear problem obtained by setting \(\varepsilon = 0\) in [eq:nls], namely \[\begin{cases} \mathrm{i}\partial_t u + \Delta u = 0,\\ u(0,\boldsymbol{x}) = \displaystyle \sum_{\boldsymbol{n} \in \mathbb{Z}^\nu} c(\boldsymbol{n})\, g_{\boldsymbol{n}}\, e^{\mathrm{i}\langle \boldsymbol{n}\boldsymbol{\Omega}, \boldsymbol{x} \rangle}. \end{cases}\] The solution is given explicitly by random Fourier series \[\label{eq:linearsolution} u_{\mathrm{linear}}(t,\boldsymbol{x}) = \sum_{\boldsymbol{n} \in \mathbb{Z}^\nu} c(\boldsymbol{n})\, g_{\boldsymbol{n}}\, e^{-\mathrm{i} t Q(\boldsymbol{n})}\, e^{\mathrm{i}\langle \boldsymbol{n}\boldsymbol{\Omega}, \boldsymbol{x} \rangle},\tag{14}\] where the dispersion relation (neglecting the minus) is \[Q(\boldsymbol{n}) = \sum_{j=1}^d \big\langle n_j,\omega_j \big\rangle^2, \qquad \boldsymbol{n}=(n_1,\cdots,n_d)\in\mathbb{Z}^{\nu_1}\times\cdots\times\mathbb{Z}^{\nu_d}.\]

Remark 7. For the proof of 2, we obey 1.

2.1 Distribution of the Linear Solution↩︎

In this subsection, we study the distribution of the linear solution and its modulus squared.

Since \(\{g_{\boldsymbol{n}}\}_{\boldsymbol{n} \in \mathbb{Z}^\nu}\) are i.i.d.standard complex Gaussian random variables, the linear evolution preserves Gaussianity. In particular, \(u_{\mathrm{linear}}(t,\boldsymbol{x})\) is a complex Gaussian random variable for every \((t,\boldsymbol{x}) \in \mathbb{R} \times \mathbb{R}^d\). Moreover, it is centered, i.e., \[\mathbb{E}\big[u_{\mathrm{linear}}(t,\boldsymbol{x})\big]=0.\] In addition, it follows from \(\mathbb{E}[g_{\boldsymbol{n}} g_{\boldsymbol{m}}] = 0\) and \(\mathbb{E}[g_{\boldsymbol{n}} \overline{g_{\boldsymbol{m}}}] = \delta_{\boldsymbol{n},\boldsymbol{m}}\) that \[\begin{align} \mathbb{E}\big[ (u_{\mathrm{linear}}(t, \boldsymbol{x}))^2 \big] &= \mathbb{E} \Bigg[ \sum_{\boldsymbol{n}, \boldsymbol{m} \in \mathbb{Z}^\nu} c(\boldsymbol{n}) c(\boldsymbol{m}) \, g_{\boldsymbol{n}} g_{\boldsymbol{m}} \, e^{-\mathrm{i} t (Q(\boldsymbol{n})+Q(\boldsymbol{m}))} e^{\mathrm{i} \langle (\boldsymbol{n}+\boldsymbol{m})\boldsymbol{\Omega}, \boldsymbol{x} \rangle} \Bigg] \nonumber \\ &= \sum_{\boldsymbol{n}, \boldsymbol{m} \in \mathbb{Z}^\nu} c(\boldsymbol{n}) c(\boldsymbol{m}) \, \mathbb{E}\big[g_{\boldsymbol{n}} g_{\boldsymbol{m}}\big] \, e^{-\mathrm{i} t (Q(\boldsymbol{n})+Q(\boldsymbol{m}))} e^{\mathrm{i} \langle (\boldsymbol{n}+\boldsymbol{m})\boldsymbol{\Omega}, \boldsymbol{x} \rangle} \nonumber \\ &=0, \end{align}\] and \[\begin{align} \mathbb{E}\big[ |u_{\mathrm{linear}}(t, \boldsymbol{x})|^2 \big] &= \mathbb{E} \Bigg[ \sum_{\boldsymbol{n}, \boldsymbol{m} \in \mathbb{Z}^\nu} c(\boldsymbol{n}) \overline{c(\boldsymbol{m})} \, g_{\boldsymbol{n}} \overline{g_{\boldsymbol{m}}} \, e^{-\mathrm{i} t (Q(\boldsymbol{n})-Q(\boldsymbol{m}))} e^{\mathrm{i} \langle (\boldsymbol{n}-\boldsymbol{m})\boldsymbol{\Omega}, \boldsymbol{x} \rangle} \Bigg] \nonumber \\ &= \sum_{\boldsymbol{n}, \boldsymbol{m} \in \mathbb{Z}^\nu} c(\boldsymbol{n}) \overline{c(\boldsymbol{m})} \, \mathbb{E}\big[g_{\boldsymbol{n}} \overline{g_{\boldsymbol{m}}}\big] \, e^{-\mathrm{i} t (Q(\boldsymbol{n})-Q(\boldsymbol{m}))} e^{\mathrm{i} \langle (\boldsymbol{n}-\boldsymbol{m})\boldsymbol{\Omega}, \boldsymbol{x} \rangle} \nonumber \\ &= \sum_{\boldsymbol{n} \in \mathbb{Z}^\nu} |c(\boldsymbol{n})|^2\nonumber\\ &\triangleq 2\sigma^2. \end{align}\]

Consequently, for each \((t,\boldsymbol{x}) \in \mathbb{R} \times \mathbb{R}^d\), the linear solution \(u_{\mathrm{linear}}(t,\boldsymbol{x})\) is distributed as a centered complex Gaussian \[u_{\mathrm{linear}}(t,\boldsymbol{x}) \sim \mathcal{N}_{\mathbb{C}}(0,2\sigma^2),\] with variance independent of both time and space.

By the properties of complex Gaussian random variables, the normalized modulus squared follows a chi-squared distribution with two degrees of freedom, which is equivalent to an exponential distribution with rate parameter \(1/2\). That is, \[\sigma^{-2} |u_{\mathrm{linear}}(t,\boldsymbol{x})|^2\sim\chi_2^2=\mathscr E\left(\frac{1}{2}\right).\] Equivalently, \(|u(t,\boldsymbol{x})|^2\) is exponentially distributed with rate parameter \(\frac{1}{2\sigma^2}\), i.e. \[|u_{\mathrm{linear}}(t,\boldsymbol{x})|^2 \sim \mathscr{E}\!\left(\frac{1}{2\sigma^2}\right),\] with probability density function \[f(x) = \frac{1}{2\sigma^2}\, e^{-\frac{x}{2\sigma^2}} \boldsymbol{1}_{[0,\infty)}(x).\]

2.2 Lower Bound↩︎

Recall that \(u_{\mathrm{linear}}(t,\boldsymbol{x})\) is a centered complex Gaussian random field with variance \(2\sigma^2\), invariant in both time and space. Consequently, for every fixed \((t,\boldsymbol{x})\in\mathbb{R}\times\mathbb{R}^d\), the random variable \(|u_{\mathrm{linear}}(t,\boldsymbol{x})|^2\) follows an exponential distribution. This enables us to obtain a lower bound through the analysis of its pointwise tail probability, which we carry out in this subsection.

To this end, for any fixed \((t,\boldsymbol{x}) \in \mathbb{R} \times \mathbb{R}^d\), we define the event \[\mathscr{A}^{(0)}_\varepsilon = \left\{ |u_{\mathrm{linear}}(t,\boldsymbol{x})| > z_0\,\varepsilon^{-1/2} \right\}.\]

Since the supremum over the spatial domain dominates any pointwise value, we have \(\mathscr{A}^{(0)}_\varepsilon \subset \mathscr{A}_\varepsilon^\infty\). Consequently, \[\begin{align} \mathbb{P}(\mathscr{A}_\varepsilon^\infty) &\geq \mathbb{P}(\mathscr{A}^{(0)}_\varepsilon) \\ &= \mathbb{P}\big(|u_{\mathrm{linear}}(t,\boldsymbol{x})|^2 \geq z_0^2 \varepsilon^{-1}\big) \\ &= \int_{z_0^2 \varepsilon^{-1}}^{\infty} \frac{1}{2\sigma^2} e^{-\frac{x}{2\sigma^2}}\, dx \\ &= \exp\!\left(-\frac{z_0^2}{2\sigma^2}\varepsilon^{-1}\right). \end{align}\]

Taking logarithms and multiplying by \(\varepsilon\), we obtain \[\varepsilon \log \mathbb{P}(\mathscr{A}_\varepsilon^\infty) \geq -\frac{z_0^2}{2\sigma^2}.\] Passing to the limit \(\varepsilon \to 0^+\) yields \[\liminf_{\varepsilon \to 0^+} \varepsilon \log \mathbb{P}(\mathscr{A}_\varepsilon^\infty) \geq -\frac{z_0^2}{2\sigma^2}.\]

Recalling that \(2\sigma^2 = \sum_{\boldsymbol{n} \in \mathbb{Z}^\nu} |c(\boldsymbol{n})|^2\), we obtain \[\label{eq:lower} \liminf_{\varepsilon \to 0^+} \varepsilon \log \mathbb{P}(\mathscr{A}_\varepsilon^\infty) \geq -\frac{z_0^2}{\sum_{\boldsymbol{n} \in \mathbb{Z}^\nu} |c(\boldsymbol{n})|^2}.\tag{16}\]

2.3 Upper Bound↩︎

In this subsection, we establish an upper bound for the probability of the rogue wave event \(\mathscr{A}_\varepsilon^\infty\). In contrast to the pointwise argument used for the lower bound, the proof of the upper bound relies on a global control of the random field, obtained via polar coordinates, truncation , Hölder’s inequality, and standard probabilistic estimates including Chernoff bound.

2.3.1 Polar Coordinates of the Random Component↩︎

Let \(g_{\boldsymbol{n}} = r_{\boldsymbol{n}} e^{\mathrm{i}\theta_{\boldsymbol{n}}}\), where \(r_{\boldsymbol{n}} \ge 0\) and \(\theta_{\boldsymbol{n}}\in[0,2\pi)\) are independent, with \(2r_{\boldsymbol{n}}^2 \overset{\mathrm{iid}}{\sim} \mathscr{E}(1/2)\) and \(\theta_{\boldsymbol{n}} \overset{\mathrm{iid}}{\sim} \mathscr{U}[0,2\pi]\). Substituting this polar representation into 14 , we obtain \[\begin{align} \label{usolution} u_{\mathrm{linear}}(t,\boldsymbol{x}) = \sum_{\boldsymbol{n}\in\mathbb{Z}^\nu} c(\boldsymbol{n})\, r_{\boldsymbol{n}}\, e^{\mathrm{i}\phi_{\boldsymbol{n}}(t,\boldsymbol{x})}, \end{align}\tag{17}\] where the phase \[\phi_{\boldsymbol{n}}(t,\boldsymbol{x}) = \theta_{\boldsymbol{n}} - Q(\boldsymbol{n})t + \langle \boldsymbol{n}\boldsymbol{\Omega}, \boldsymbol{x} \rangle.\]

To control the supremum of the field, we observe that \[\begin{align} \sup_{\boldsymbol{x}\in\mathbb{R}^d} |u_{\mathrm{linear}}(t,\boldsymbol{x})|^2 &= \sup_{\boldsymbol{x}\in\mathbb{R}^d} \operatorname{Re}\!\left(u_{\mathrm{linear}}(t,\boldsymbol{x})\,\overline{u_{\mathrm{linear}}(t,\boldsymbol{x})}\right) \nonumber\\ &= \sup_{\boldsymbol{x}\in\mathbb{R}^d} \sum_{\boldsymbol{n},\boldsymbol{m}\in\mathbb{Z}^\nu} r_{\boldsymbol{n}} r_{\boldsymbol{m}} \operatorname{Re}\!\left( c(\boldsymbol{n})\,\overline{c(\boldsymbol{m})}\, e^{\mathrm{i}(\phi_{\boldsymbol{n}}(t,\boldsymbol{x})-\phi_{\boldsymbol{m}}(t,\boldsymbol{x}))} \right) \nonumber\\ &\le \sum_{\boldsymbol{n},\boldsymbol{m}\in\mathbb{Z}^\nu} r_{\boldsymbol{n}} r_{\boldsymbol{m}}\, |c(\boldsymbol{n})|\, |c(\boldsymbol{m})| \nonumber\\ &= \left(\sum_{\boldsymbol{n}\in\mathbb{Z}^\nu} |c(\boldsymbol{n})|\, r_{\boldsymbol{n}}\right)^2. \end{align}\]

Consequently, we obtain the uniform bound \[\begin{align} \label{eq:cr} \sup_{\boldsymbol{x}\in\mathbb{R}^d} |u_{\mathrm{linear}}(t,\boldsymbol{x})| \le \sum_{\boldsymbol{n}\in\mathbb{Z}^\nu} |c(\boldsymbol{n})|\, r_{\boldsymbol{n}}, \end{align}\tag{18}\] in which we have gotten rid of the supremum on \(\mathbb{R}^d\).

2.3.2 Truncation↩︎

To study the probability of \(\mathscr A_\varepsilon^{(1)}\), we introduce the truncation parameter \[\begin{align} \label{eq:N} N=\varepsilon^{-\frac{1}{\nu}+\mu_1}, \qquad 0<\mu_1\ll\frac{1}{\nu}. \end{align}\tag{20}\]

We define the truncated lattice \(\Lambda_N\subset\mathbb{Z}^\nu\) by \[\begin{align} \Lambda_N &=\left\{\boldsymbol{n}\in\mathbb{Z}^\nu:\;|\boldsymbol{n}|\le N\right\} \\ &=\left\{ (n_1,\ldots,n_d)\in\prod_{j=1}^{d}\mathbb{Z}^{\nu_j} :\;|n_j|\le N,\quad j=1,\ldots,d \right\} \\ &=\left\{ (n_{1,1},\ldots,n_{1,\nu_1};\,\ldots;\,n_{d,1},\ldots,n_{d,\nu_d}) \in\prod_{j=1}^{d}\mathbb{Z}^{\nu_j} :\;|n_{j,j'}|\le N,\; j'=1,\ldots,\nu_j,\; j=1,\ldots,d \right\}. \end{align}\]

Using the truncation set \(\Lambda_N\), we decompose the right-hand side of 18 into a principal part \(\mathscr S_N\) and a remainder term \(\mathscr R_N\): \[\label{eq:sr} \sum_{\boldsymbol{n}\in\mathbb{Z}^\nu} |c(\boldsymbol{n})|\,r_{\boldsymbol{n}} = \mathscr S_N+\mathscr R_N,\tag{21}\] where \[\mathscr S_N = \sum_{\boldsymbol{n}\in\Lambda_N} |c(\boldsymbol{n})|\,r_{\boldsymbol{n}}, \qquad \mathscr R_N = \sum_{\boldsymbol{n}\notin\Lambda_N} |c(\boldsymbol{n})|\,r_{\boldsymbol{n}}.\]

2.3.3 Analysis of \(\mathscr R_N\)↩︎

2.3.4 Analysis of \(\mathscr S_N\).↩︎

Here we analyze the principal part \(\mathscr{S}_N\), which provides the dominant contribution to the upper bound. The analysis reduces to studying a sum of independent random variables after separating deterministic and random components.

Clearly, \[\mathscr A^{(2)}_\varepsilon=\left\{\mathscr S_N^2>(z_0-C\varepsilon^{\frac{\mu_2}{2}})^2 \varepsilon^{-1}\right\}.\] Applying Hölder’s inequality, we obtain \[\begin{align} \mathscr S_N^2 &\leq \sum_{\boldsymbol{n}\in\Lambda_N}\frac{1}{2}|c(\boldsymbol{n})|^2 \cdot \sum_{\boldsymbol{n}\in\Lambda_N}2r_{\boldsymbol{n}}^2 \nonumber\\ &\leq \left(\sum_{\boldsymbol{n}\in\mathbb{Z}^\nu}\frac{1}{2}|c(\boldsymbol{n})|^2\right)\cdot \left(\sum_{\boldsymbol{n}\in\Lambda_N}2r_{\boldsymbol{n}}^2\right). \end{align}\]

2.3.5 Putting Everything Together↩︎

It follows from 19 , 22 , 24 , 25 , 27 and the law of total probability that \[\begin{align} \mathbb{P}(\mathscr A_\varepsilon^\infty)\le\mathbb{P}(\mathscr A_\varepsilon^{(1)}) &=\mathbb{P}(\mathscr A_\varepsilon^{(1)}\cap\widetilde{\Omega})+ \mathbb{P}(\mathscr A_\varepsilon^{(1)}\cap\widetilde{\Omega}^c)\\ &\le\mathbb{P}(\mathscr A_\varepsilon^{(2)})+\mathbb{P}(\widetilde{\Omega}^c)\\ &\le\mathbb{P}(\mathscr A_\varepsilon^{(3)})+\mathbb{P}(\Omega_\delta)\\ &=\mathbb{P}(\mathscr A_\varepsilon^{(3)})\left(1+\frac{\mathbb{P}(\Omega_\delta)}{\mathbb{P}(\mathscr A_\varepsilon^{(3)})}\right). \end{align}\]

Taking logarithms and multiplying by \(\varepsilon\), we have \[\begin{align} \label{ee} \varepsilon\log\mathbb{P}(\mathscr A_\varepsilon^\infty)\leq\varepsilon\log\mathbb{P}(\mathscr A^{(3)}_\varepsilon)+\varepsilon\log\left(1+\frac{\mathbb{P}(\Omega_\delta)}{\mathbb{P}(\mathscr A_\varepsilon^{(3)})}\right). \end{align}\tag{28}\] The first term satisfies a large deviation principle (see 2), namely \[\mathbb{P}(\mathscr A_\varepsilon^{(3)}) = \exp\!\left(-\frac{\mathfrak I_0}{2\varepsilon} + o(\varepsilon^{-1})\right).\] The second term is controlled by 2, namely \[\mathbb{P}(\Omega_\delta) \lesssim \exp(-\delta^{-1}),\quad \delta\sim\varepsilon^{1+\eta}<\varepsilon,\] where \(\eta\) is specified in 1. Thus, \[\lim_{\varepsilon \to 0^+} \varepsilon \log\!\left(1 + \frac{\mathbb{P}(\Omega_\delta)}{\mathbb{P}(\mathscr A_\varepsilon^{(3)})}\right) = 0.\]

Taking the upper limit with respect to \(\varepsilon\to0^+\) on both sides of 28 , we have \[\begin{align} \limsup_{\varepsilon \to 0^+} \varepsilon \log \mathbb{P}(\mathscr A_\varepsilon^\infty) &\le -\frac{z_0^2}{\sum_{\boldsymbol{n}\in\mathbb{Z}^\nu} |c(\boldsymbol{n})|^2}, \end{align}\] which agrees with the right-hand side of 16 . This completes the proof of 2.

3 Well-Posedness↩︎

In 2, we established the linear LDP, which serves as a baseline for the nonlinear analysis. Before turning to the nonlinear case, we first derive in 3 a suitable well-posedness result that captures the relevant time scale, along with the necessary estimates.

The proof of 3 is based on the strategy developed in [29]–[31]. We outline the main steps, emphasizing the key modifications.

3.1 The Lattice System and Picard Iteration↩︎

It follows from the orthogonality of \(\{e^{\mathrm{i}\langle\boldsymbol{n}\Omega,\boldsymbol{x}\rangle}\}_{\boldsymbol{n}\in\mathbb{Z}^\nu}\) (see [31]) that, in Fourier space, [eq:nls] reads \[\begin{align} \label{eq:fourier95nls} \partial_t{c}(t,\boldsymbol{n}) = -\mathrm{i} Q(\boldsymbol{n}){c}(t,\boldsymbol{n}) + \mathrm{i}{\varepsilon^\alpha} \sum_{\substack{\boldsymbol{n}^j \in \mathbb{Z}^\nu,\, j=1,2,3\\ \boldsymbol{n}^1 - \boldsymbol{n}^2 + \boldsymbol{n}^3 = \boldsymbol{n}}} \prod_{j=1}^3 \bigl\{{c}(t,\boldsymbol{n}^j)\bigr\}^{\ast^{[j-1]}}. \end{align}\tag{31}\]

The initial condition in Fourier space is given by \[\begin{align} \label{eq:initialdata95fourier} {c}(0,\boldsymbol{n})={c}(\boldsymbol{n})g_{\boldsymbol{n}}. \end{align}\tag{32}\]

For each \(\boldsymbol{n} \in \mathbb{Z}^\nu\), the Duhamel formulation reads \[\begin{align} \label{eq:duhamel95c} {c}(t,\boldsymbol{n}) = e^{-\mathrm{i}Q(\boldsymbol{n})t}{c}(\boldsymbol{n})g_{\boldsymbol{n}} + \mathrm{i}{\varepsilon^\alpha} \int_0^t e^{-\mathrm{i}Q(\boldsymbol{n})(t-s)} \sum_{\substack{\boldsymbol{n}^j \in \mathbb{Z}^\nu,\, j=1,2,3\\ \boldsymbol{n}^1 - \boldsymbol{n}^2 + \boldsymbol{n}^3 = \boldsymbol{n}}} \prod_{j=1}^{3} \bigl\{{c}(s,\boldsymbol{n}^j)\bigr\}^{\ast^{[j-1]}} \,\mathrm{d}s. \end{align}\tag{33}\]

Define the Picard iteration \(\{{c}_k(t,\boldsymbol{n})\}_{k\ge0}\) by \[\begin{align} \label{eq:initialguess} {c}_0(t,\boldsymbol{n}) = e^{-\mathrm{i}Q(\boldsymbol{n})t}{c}(\boldsymbol{n})g_{\boldsymbol{n}}, \end{align}\tag{34}\] and for \(k\ge1\), \[\begin{align} \label{eq:picard} {c}_k(t,\boldsymbol{n}) = {c}_0(t,\boldsymbol{n}) + \mathrm{i}{\varepsilon^\alpha} \int_0^t e^{-\mathrm{i}Q(\boldsymbol{n})(t-s)} \sum_{\substack{\boldsymbol{n}^j \in \mathbb{Z}^\nu,\, j=1,2,3\\ \boldsymbol{n}^1 - \boldsymbol{n}^2 + \boldsymbol{n}^3 = \boldsymbol{n}}} \prod_{j=1}^{3} \bigl\{{c}_{k-1}(s,\boldsymbol{n}^j)\bigr\}^{\ast^{[j-1]}} \,\mathrm{d}s. \end{align}\tag{35}\]

3.2 Tree Structure of the Picard Sequence↩︎

In this subsection, we represent the Picard iteration using tree expansions, in order to systematically encode the combinatorial structure of the Duhamel terms. This representation allows us to track the multilinear interactions arising from the nonlinear evolution; see 6.

We begin by illustrating the decomposition of the linear and nonlinear contributions (see 2 and 3). We then briefly recall the combinatorial formalism developed in [30]–[32], which provides the basis for the tree representation.

3.3 Uniform-in-Time Polynomial Decay of the Picard Sequence↩︎

Let \(\boldsymbol{n}\in\mathbb{Z}^\nu\) be fixed. We also expand it as \[(n_1,\cdots,n_d)\in\mathbb{Z}^{\nu_1}\times\cdots\times\mathbb{Z}^{\nu_d},\] and further write \[(n_{1,1},\cdots,n_{1,\nu_1};\cdots;n_{d,1},\cdots,n_{d,\nu_d})\in\underbrace{\mathbb{Z}\times\cdots\times\mathbb{Z}}_{\nu_1}\times\cdots\times\underbrace{\mathbb{Z}\times\cdots\times\mathbb{Z}}_{\nu_d}.\]

For \(\gamma^{(k)}=0\in\Gamma^{(k)}\) with \(k\geq1\) (see 2), we set \(\boldsymbol{m}^1=\boldsymbol{n}\), that is, \((m^1_1,\cdots,m^1_d)=(n_1,\cdots,n_d)\), or equivalently, \[\left(m^1_{1,1},\cdots,m^1_{1,\nu_1}; \cdots; m^1_{d,1},\cdots, m^1_{d,\nu_d}\right) = \left(n_{1,1},\cdots,n_{1,\nu_1}; \cdots; n_{d,1},\cdots, n_{d,\nu_d}\right).\]

For \(\gamma^{(k)}=(\gamma_1^{(k-1)},\gamma_2^{(k-1)},\gamma_3^{(k-1)})\in\Gamma^{(k)}\setminus\{0\}\), we write \(\boldsymbol{n}=\boldsymbol{n}^1-\boldsymbol{n}^2+\boldsymbol{n}^3\in\mathbb{Z}^\nu\) (see 3), and set \[\begin{align} \mathbb{Z}^\nu\ni\boldsymbol{m}^j= \begin{cases} \boldsymbol{n}^{1,j}, & 1\leq j\leq 2\sigma(\gamma_1^{(k-1)});\\[4pt] \boldsymbol{n}^{2,\,j-2\sigma(\gamma_1^{(k-1)})}, & 2\sigma(\gamma_1^{(k-1)})+1\le j\le 2\sigma(\gamma_1^{(k-1)})+2\sigma(\gamma_2^{(k-1)});\\[4pt] \boldsymbol{n}^{3,\,j-2\sigma(\gamma_1^{(k-1)})-2\sigma(\gamma_2^{(k-1)})}, & 2\sigma(\gamma_1^{(k-1)})+2\sigma(\gamma_2^{(k-1)})+1\le j\le 2\sigma(\gamma^{(k)}). \end{cases} \end{align}\] Equivalently, for each component \(s=1,\dots,d\), we have \[\begin{align} \mathbb{Z}^{\nu_{s}}\ni m_{s}^j= \begin{cases} n_{s}^{1,j}, & 1\leq j\leq 2\sigma(\gamma_1^{(k-1)});\\[4pt] n_{s}^{2,\,j-2\sigma(\gamma_1^{(k-1)})}, & 2\sigma(\gamma_1^{(k-1)})+1\le j\le 2\sigma(\gamma_1^{(k-1)})+2\sigma(\gamma_2^{(k-1)});\\[4pt] n_{s}^{3,\,j-2\sigma(\gamma_1^{(k-1)})-2\sigma(\gamma_2^{(k-1)})}, & 2\sigma(\gamma_1^{(k-1)})+2\sigma(\gamma_2^{(k-1)})+1\le j\le 2\sigma(\gamma^{(k)}), \end{cases} \end{align}\] where \(m_{s}^j=\left(m^j_{s,1},\cdots,m^j_{s,\nu_{s}}\right)\in\mathbb{Z}^{\nu_{s}}\).

Thus \[\widetilde{\spadesuit}^{(k)}_{x_s}=(m_s^j)_{1\le j\le2\sigma(\gamma^{(k)})},\qquad 1\le s\le d,\quad k\geq1.\]

4 Nonlinear LDP↩︎

In this section, we combine the linear LDP (1) with the well-posedness result (3) to establish the nonlinear LDP (1). The main idea is to transfer the large deviations estimates from the linearized equation to the full nonlinear dynamics via a suitable perturbative argument. In particular, the well-posedness theory guarantees that the nonlinear solution can be constructed globally in the regime under consideration, which allows us to compare it with the corresponding linear evolution and control the nonlinear error terms; see 4.

4.1 Duhamel Estimates↩︎

4.2 Lower bound↩︎

Set \[\begin{align} \mathscr A_\varepsilon^- &= \left\{ \| u_{\text{linear}} \|_{L^\infty_x(\mathbb{R}^d)} - \| u - u_{\text{linear}} \|_{L^\infty_x(\mathbb{R}^d)} > z_0 \, \varepsilon^{-\frac{1}{2}} \right\},\\ \mathscr B_\varepsilon^- &= \left\{ \| u_{\text{linear}} \|_{L^\infty_x(\mathbb{R}^d)} > \left( z_0 + C \, \varepsilon^{\frac{\mu_3}{2}} \right) \varepsilon^{-\frac{1}{2}} \right\}. \end{align}\] Clearly, we have the following inclusion relation: \[\mathscr B_\varepsilon^-\cap\widehat{\Omega} \subset \mathscr A_\varepsilon^- \subset \mathscr A_\varepsilon^\infty.\]

By the law of total probability and 41 , we have \[\begin{align} \mathbb{P}(\mathscr A_\varepsilon^\infty) &\ge \mathbb{P}(\mathscr B_\varepsilon^-\cap\widehat{\Omega})\\ &= \mathbb{P}(\mathscr B_\varepsilon^-) -\mathbb{P}(\mathscr B_\varepsilon^-\cap\widehat{\Omega}^{\,c})\\ &\ge \mathbb{P}(\mathscr B_\varepsilon^-)-\mathbb{P}(\widehat{\Omega}^{\,c})\\ &\ge \mathbb{P}(\mathscr B_\varepsilon^-)-\mathbb{P}(\Omega_\delta)\\ &=\mathbb{P}(\mathscr B_\varepsilon^-)\left(1-\frac{\mathbb{P}(\Omega_\delta)}{\mathbb{P}(\mathscr B_\varepsilon^-)}\right). \end{align}\] Taking logarithms and multiplying by \(\varepsilon\), we have \[\begin{align} \label{w2} \varepsilon\log\mathbb{P}(\mathscr A_\varepsilon^\infty)\geq \varepsilon\log\mathbb{P}(\mathscr B_\varepsilon^-) +\varepsilon\log\left(1-\frac{\mathbb{P}(\Omega_\delta)}{\mathbb{P}(\mathscr B_\varepsilon^-)}\right). \end{align}\tag{42}\]

Applying 2, we deduce that \[\begin{align} \lim_{\varepsilon \to 0^+} \varepsilon \log\mathbb{P} \left(\mathscr B_\varepsilon^- \right) &= -\frac{z_0^2}{\sum_{\boldsymbol{n}\in\mathbb{Z}^\nu}| c(\boldsymbol{n})|^2}. \end{align}\]

Similar to the analysis in 2.3.5, \[\lim_{\varepsilon\to0}\varepsilon\log\left(1-\frac{\mathbb{P}(\Omega_\delta)}{\mathbb{P}(\mathscr B_\varepsilon^-)}\right)=0.\]

Therefore \[\begin{align} \label{eqlower} \liminf_{\varepsilon\to0^+}\varepsilon\log\mathbb{P}(\mathscr A_\varepsilon^\infty) \ge-\frac{z_0^2}{\sum_{\boldsymbol{n}\in\mathbb{Z}^\nu}| c(\boldsymbol{n})|^2}. \end{align}\tag{43}\]

4.3 Upper bound↩︎

Set \[\begin{align} \mathscr A_\varepsilon^+ &= \left\{ \| u_{\text{linear}} \|_{L^\infty_x(\mathbb{R}^d)} + \| u - u_{\text{linear}} \|_{L^\infty_x(\mathbb{R}^d)} > z_0 \, \varepsilon^{-\frac{1}{2}} \right\},\\ \mathscr B_\varepsilon^+ &= \left\{ \| u_{\text{linear}} \|_{L^\infty_x(\mathbb{R}^d)} > \left( z_0 - C \, \varepsilon^{\frac{\mu_3}{2}} \right) \varepsilon^{-\frac{1}{2}} \right\}. \end{align}\] Clearly, we have the following inclusion relation: \[\mathscr A_\varepsilon^\infty \subset \mathscr A_\varepsilon^+,\quad \mathscr A_\varepsilon^+\cap\widehat{\Omega} \subset \mathscr B_\varepsilon^+.\]

It follows from the law of total probability and 41 that \[\begin{align} \mathbb{P}(\mathscr A_\varepsilon^\infty) &\le \mathbb{P}(\mathscr A_\varepsilon^+)\\ &=\mathbb{P}(\mathscr A_\varepsilon^+\cap\widehat{\Omega}) +\mathbb{P}(\mathscr A_\varepsilon^+\cap\widehat{\Omega}^c)\\ &\le \mathbb{P}(\mathscr B_\varepsilon^+)+\mathbb{P}(\widehat{\Omega}^c)\\ &\le \mathbb{P}(\mathscr B_\varepsilon^+)+\mathbb{P}(\Omega_\delta)\\ &=\mathbb{P}(\mathscr B_\varepsilon^+)\left(1+\frac{\mathbb{P}(\Omega_\delta)}{\mathbb{P}(\mathscr B_\varepsilon^+)}\right). \end{align}\] Taking logarithms and multiplying by \(\varepsilon\), we have \[\begin{align} \label{w3} \varepsilon\log\mathbb{P}(\mathscr A_\varepsilon^\infty)\le \varepsilon\log\mathbb{P}(\mathscr B_\varepsilon^+) +\varepsilon\log\left(1+\frac{\mathbb{P}(\Omega_\delta)}{\mathbb{P}(\mathscr B_\varepsilon^+)}\right). \end{align}\tag{44}\]

Applying 2, we deduce that \[\begin{align} \lim_{\varepsilon \to 0^+} \varepsilon \log \mathbb{P}\left(\mathscr B_\varepsilon^+ \right) &= -\frac{z_0^2}{\sum_{\boldsymbol{n}\in\mathbb{Z}^\nu}| c(\boldsymbol{n})|^2}. \end{align}\]

Analogously, \[\lim_{\varepsilon\to0}\varepsilon\log\left(1+\frac{\mathbb{P}(\Omega_\delta)}{\mathbb{P}(\mathscr B_\varepsilon^+)}\right)=0.\]

Hence we obtain \[\begin{align} \label{equpper} \limsup_{\varepsilon\to0^+}\varepsilon\log\mathbb{P}(\mathscr A_\varepsilon^\infty) \le-\frac{z_0^2}{\sum_{\boldsymbol{n}\in\mathbb{Z}^\nu}| c(\boldsymbol{n})|^2}. \end{align}\tag{45}\]

Combining 43 and 45 yields that \[\begin{align} \lim_{\varepsilon\to0^+}\varepsilon\log\mathbb{P}\Big( \|u(t)\|_{L_{\boldsymbol{x}}^\infty(\mathbb{R}^d)} > z_0 \varepsilon^{-\frac{1}{2}} \Big) =-\frac{z_0^2}{\sum_{\boldsymbol{n}\in\mathbb{Z}^\nu}|c(\boldsymbol{n})|^2},\quad t\sim\varepsilon^{-\alpha+1+\frac{3}{2}\eta+\frac{\mu_3}{2}}. \end{align}\] This completes the proof of 1.

5 Probability and Statistics↩︎

In this section, we review some fundamental notation and concepts from probability and statistics.

5.1 Real-Valued Random Variables↩︎

Let the triple \((\Omega, \mathcal{F}, \mathbb{P})\) be a probability space. The set \(\Omega\) denotes the sample space, comprising all possible outcomes, and \(\omega \in \Omega\) represents an individual outcome (sample point). The collection \(\mathcal{F}\) is a \(\sigma\)-algebra on \(\Omega\), referred to as the event space, whose elements are subsets of \(\Omega\); these events are typically denoted by uppercase letters such as \(A\), \(B\), \(C\). The function \(\mathbb{P} : \mathcal{F} \to [0,1]\) is a probability measure, assigning to each event \(A \in \mathcal{F}\) its probability \(\mathbb{P}(A)\).

Let \(\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}\) be a field. A \(\mathbb{K}\)-valued random variable \(X\) is a measurable function from the sample space to \(\mathbb{K}\), that is, \(X : \Omega \to \mathbb{K}\), \(\omega \mapsto X(\omega)\).

We now focus on real-valued random variables, i.e., the case \(\mathbb{K} = \mathbb{R}\). Events of the form \(\{\omega \in \Omega : X(\omega) \in D \subset \mathbb{R}\} := A\) are commonly abbreviated as \(\{X \in D\}\). Furthermore, \(\mathbb{P}(A) = \mathbb{P}(\{X \in D\}) \triangleq \mathbb{P}(X \in D)\).

The cumulative distribution function (CDF) of a random variable \(X\), denoted by \(F_X\), is defined as \(F_X(x) = \mathbb{P}(X \leq x)\) for all \(x \in \mathbb{R}\).

Under standard regularity conditions, such as the differentiability of \(F_X\) with respect to \(x\), the probability density function (PDF) of \(X\), denoted \(f_X\), is defined as the derivative \[f_X(x) = \frac{{\rm d}}{{\rm d}x} F_X(x).\]

The expectation, or mean, of a random variable \(X\), denoted \(\mathbb{E}[X]\), is the weighted average of \(X(\omega)\) (or \(X\)) using the probability measure \({\rm d}\mathbb{P}(\omega)\) (or PDF \(f_X\)) as weights, that is, \[\begin{align} \mathbb{E}[X] &= \int_\Omega X(\omega)\,{\rm d}\mathbb{P}(\omega) \quad (\text{using the probability measure in abstract space}), \\ &= \int_{\mathbb{R}} x\,{\rm d}F_X(x) \quad (\text{using the CDF via Stieltjes integral}), \\ &= \int_{\mathbb{R}} x f_X(x)\,{\rm d}x \quad (\text{using the PDF in the absolutely continuous case}). \end{align}\] We also denote this by \(\mu_X \triangleq \mu\).

The expectation satisfies linearity, that is, \[\mathbb{E}[a_1X_1 + a_2X_2] = a_1\mathbb{E}[X_1] + a_2\mathbb{E}[X_2], \quad \forall a_1,a_2 \in \mathbb{R}.\]

The variance of a random variable \(X\), denoted \(\mathbb{V}(X)\), is defined as the expected value of \((X-\mu_X)^2\), that is, \[\mathbb{V}(X) = \mathbb{E}[(X-\mu_X)^2].\] We also denote this by \(\sigma_X^2 \triangleq \sigma^2\). By linearity of expectation, \[\mathbb{V}(X) = \mathbb{E}[X^2] - \mu_X^2.\] Furthermore, \[\mathbb{V}(aX + b) = a^2 \mathbb{V}(X), \quad \forall a \in \mathbb{R}.\]

The characteristic function (CF) of \(X\), denoted \(\phi_X\), is defined by \[\phi_X(t) = \mathbb{E}[e^{{\rm i} t X}], \quad t \in \mathbb{R}.\] It should be noted that characteristic functions uniquely determine the distribution: \(\phi_X = \phi_Y\) if and only if \(X\) and \(Y\) have the same distribution. This follows from the Lévy continuity theorem; see [33].

5.2 Several Important Distributions↩︎

In this subsection, we briefly introduce several important classical distributions relevant to studying rogue wave mechanisms from a probabilistic perspective. These distributions are repeatedly applied in the calculation of LDP for rogue waves.

5.2.1 Gaussian Distribution↩︎

We say that a random variable \(X\) follows the Gaussian/normal distribution with mean \(\mu\) and variance \(\sigma^2\) if its PDF is \[f_X(x;\mu,\sigma) = \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}}, \quad x \in \mathbb{R}.\] The corresponding mean, variance and characteristic function are, respectively, \[\mathbb{E}[X] = \mu, \quad \mathbb{V}(X) = \sigma^2, \quad \phi_X(t) = e^{{\rm i}\mu t - \frac{1}{2}\sigma^2 t^2}.\]

Here and throughout, we write \(X \sim \mathscr{N}_\mathbb{R}(\mu,\sigma^2)\) to denote a real-valued random variable following this distribution.

If \(X \sim \mathscr{N}_\mathbb{R}(\mu,\sigma^2)\), then the normalization \(Y = \frac{X-\mu}{\sigma} \sim \mathscr{N}_\mathbb{R}(0,1)\). In this case, we say that \(Y\) follows the standard Gaussian/normal distribution, with PDF \(f_Y(y) = \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}y^2}\) and CF \(\phi_Y(t) = e^{-\frac{1}{2}t^2}\).

5.2.2 Exponential Distribution↩︎

We say that a random variable \(X\) follows the exponential distribution with rate parameter \(\lambda > 0\) if its PDF is \[f_X(x;\lambda) = \lambda e^{-\lambda x} \boldsymbol{1}_{[0,\infty)}(x).\]

Throughout, we write \(X \sim \mathscr{E}(\lambda)\) to denote that \(X\) follows this distribution.

If \(X \sim \mathscr{E}(\lambda)\), then \[\mathbb{E}[X] = \frac{1}{\lambda}, \quad \mathbb{V}(X) = \frac{1}{\lambda^2},\] and \(aX \sim \mathscr{E}\bigl(\frac{\lambda}{a}\bigr)\) for \(a > 0\).

5.2.3 Chi-squared Distribution↩︎

We say that a random variable \(X\) follows the chi-squared distribution with \(n\) degrees of freedom if its PDF is \[f_X(x;n) = \frac{1}{\Gamma\left(\frac{n}{2}\right)2^{n/2}} x^{\frac{n}{2}-1} e^{-x/2} \boldsymbol{1}_{[0,\infty)}(x).\]

Throughout, we write \(X \sim \chi_n^2\) to denote this distribution.

If \(X \sim \chi_n^2\), then \[\mathbb{E}[X] = n, \quad \mathbb{V}(X) = 2n.\]

In addition, we have

• \(\chi_2^2 = \mathscr{E}\bigl(\frac{1}{2}\bigr)\).

• If \(X_1 \sim \chi_{n_1}^2\) and \(X_2 \sim \chi_{n_2}^2\) are independent, then \(X_1 + X_2 \sim \chi_{n_1+n_2}^2\).

• If \(\{X_j\}_{1\leq j\leq n}\) are i.i.d. \(\mathscr{E}(\lambda)\), then \(2\lambda\sum_{j=1}^n X_j \sim \chi_{2n}^2\).

• If \(\{X_j\}_{1\leq j\leq n}\) are i.i.d. \(\mathscr{N}_\mathbb{R}(0,1)\), then \(\sum_{j=1}^n X_j^2 \sim \chi_n^2\).

To introduce complex-valued random variables, we first review the concepts of joint distributions (see 5.3) and random independence (see 5.4), initially for two random variables. These concepts naturally generalize to any finite collection of random variables.

5.3 Joint Distribution↩︎

Let \(X\) and \(Y\) be random variables defined on a common probability space \((\Omega, \mathcal{F}, \mathbb{P})\). The joint CDF of \((X,Y)\), denoted \(F_{X,Y}\), is defined by \[F_{X,Y}(x,y) = \mathbb{P}(X \leq x, Y \leq y), \quad (x,y) \in \mathbb{R}^2.\]

Under standard regularity conditions (such as the existence of mixed partial derivatives), the joint PDF, denoted \(f_{X,Y}\), is given by \[f_{X,Y}(x,y) = \frac{\partial^2 F_{X,Y}(x,y)}{\partial x \partial y}.\]

Let \(f_X\) and \(f_Y\) denote the marginal PDFs of \(X\) and \(Y\). Then \[\begin{align} f_X(x) &= \int_{\mathbb{R}} f_{X,Y}(x,y)\,{\rm d}y = \frac{\partial F_{X,Y}(x,y)}{\partial x}\Big|_{y\to +\infty}, \\ f_Y(y) &= \int_{\mathbb{R}} f_{X,Y}(x,y)\,{\rm d}x = \frac{\partial F_{X,Y}(x,y)}{\partial y}\Big|_{x\to +\infty}. \end{align}\] Here, \(f_X\) and \(f_Y\) are called the marginal distributions.

The joint CF for the joint distribution of \((X,Y)\), denoted by \(\phi_{X,Y}\), is defined by \[\phi_{X,Y}(t_1,t_2) = \mathbb{E}[e^{{\rm i}(t_1X + t_2Y)}], \quad (t_1,t_2) \in \mathbb{R}^2.\]

5.4 Conditional Probability and Independence↩︎

Let \(A\) and \(B\) be events in the probability space \((\Omega, \mathcal{F}, \mathbb{P})\) with \(\mathbb{P}(B) > 0\). The conditional probability of \(A\) given \(B\), denoted \(\mathbb{P}(A|B)\), is defined by \[\mathbb{P}(A|B) = \frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)}.\] This yields the multiplication rule: \[\mathbb{P}(A \cap B) = \mathbb{P}(A|B) \mathbb{P}(B).\]

Events \(A\) and \(B\) are independent if \[\mathbb{P}(A|B) = \mathbb{P}(A),\] or equivalently, \[\mathbb{P}(A \cap B) = \mathbb{P}(A) \mathbb{P}(B).\] In this case, \(\mathbb{P}(B|A) = \mathbb{P}(B)\) also holds.

Let \(X\) and \(Y\) be random variables defined on a common probability space \((\Omega, \mathcal{F}, \mathbb{P})\). We say \(X\) and \(Y\) are independent if the events \(\{X \leq x\}\) and \(\{Y \leq y\}\) are independent for all \(x,y \in \mathbb{R}\). Specifically, \[\mathbb{P}(X \leq x, Y \leq y) = \mathbb{P}(X \leq x) \mathbb{P}(Y \leq y).\] That is, \[\begin{align} F_{X,Y}(x,y) &= F_X(x) F_Y(y) \quad (\text{using the CDF}), \\ f_{X,Y}(x,y) &= f_X(x) f_Y(y) \quad (\text{using the PDF}), \\ \phi_{X,Y}(t_1,t_2) &= \phi_X(t_1) \phi_Y(t_2) \quad (\text{using the CF}). \end{align}\]

5.5 Complex-Valued Random Variables↩︎

From now on, we focus on complex-valued random variables, i.e., \(\mathbb{K} = \mathbb{C}\), particularly those following complex Gaussian distributions.

Let \(Z\) be a complex random variable on the probability space \((\Omega, \mathcal{F}, \mathbb{P})\), that is, \(Z : \Omega \to \mathbb{C}\), \(\omega \mapsto Z(\omega)\).

The distribution of \(Z\) admits two natural representations. The first, based on the Cartesian coordinate system, identifies it with the joint distribution of the real and imaginary parts \(\operatorname{Re}Z\) and \(\operatorname{Im}Z\). The second, based on the polar coordinate system, identifies it with the joint distribution of the modulus \(|Z|\) and argument \(\arg Z\).

In the Cartesian approach, write \(Z(\omega) = X(\omega) + {\rm i} Y(\omega)\) for \(\omega \in \Omega\), where \(X,Y\) are real-valued random variables on \((\Omega, \mathcal{F}, \mathbb{P})\). The distribution of \(Z\) is the joint distribution of \((X,Y)\) under the partial order \(\prec\) defined by \[z_1 \prec z_2 \Longleftrightarrow \operatorname{Re} z_1 \leq \operatorname{Re} z_2 \;\text{and}\; \operatorname{Im} z_1 \leq \operatorname{Im} z_2.\] For \(z = x + {\rm i} y \in \mathbb{C}\), the CDF is \[F_Z(z) = \mathbb{P}(Z \prec z) = \mathbb{P}(X \leq x, Y \leq y) = F_{X,Y}(x,y).\] Hence the PDF is given by \[f_Z(z)=f_{X,Y}(x, y)=\frac{\partial^2F_{X,Y}(x,y)}{\partial x\partial y};\] and for \(t=t_1+{\rm i}t_2\), where \(t_1,t_2\in\mathbb{R}\), the CF is given by³ \[\phi_Z(t) = \phi_{X,Y}(t_1, t_2) = \mathbb{E} \left[ e^{\mathrm{i}(t_1 X + t_2 Y)} \right], \quad (t_1, t_2) \in \mathbb{R}^2.\]

For the expectation of \(Z = X + {\rm i} Y\), we have \[\mathbb{E}[Z] = \mathbb{E}[X] + {\rm i} \mathbb{E}[Y].\] Whenever \(\mathbb{E}[Z]\) exists, expectation and complex conjugation commute, that is, \[\mathbb{E}[\overline{Z}] = \overline{\mathbb{E}[Z]}.\]

For the variance of \(Z = X + {\rm i} Y\), it is defined by \[\mathbb{V}(Z) = \mathbb{E}[|Z - \mu_Z|^2].\] Using \(|z|^2 = z\overline{z}\), we have \[\begin{align} \mathbb{V}(Z) &= \mathbb{E}\bigl[(Z-\mu_Z)(\overline{Z}-\overline{\mu_Z})\bigr] \\ &= \mathbb{E}[|Z|^2] - |\mu_Z|^2. \end{align}\] Using \(|z|^2 = x^2 + y^2\) with \(Z-\mu_Z = (X-\mu_X) + {\rm i}(Y-\mu_Y)\), we obtain \[\begin{align} \mathbb{V}(Z) &= \mathbb{E}[|X-\mu_X|^2 + |Y-\mu_Y|^2] \\ &= \mathbb{V}(X) + \mathbb{V}(Y). \end{align}\]

In the polar approach, write \(Z(\omega) = R(\omega) e^{{\rm i} \Theta(\omega)}\) for \(\omega \in \Omega\), where \(R(\omega) > 0\), \(0 \leq \Theta(\omega) < 2\pi\), and \(R, \Theta\) are real-valued random variables on \((\Omega, \mathcal{F}, \mathbb{P})\). The distribution of \(Z\) is then identified with the joint distribution of \((R, \Theta)\).

In fact, these two approaches are equivalent via the transformation \[X = R \cos \Theta, \quad Y = R \sin \Theta.\]

5.5.1 Complex Gaussian↩︎

Here we focus on centered complex-valued Gaussian random variables, i.e., those with zero mean. Write \(Z = X + {\rm i}Y,\) where \(X \sim \mathscr{N}_\mathbb{R}(0,\sigma_1^2)\) and \(Y \sim \mathscr{N}_\mathbb{R}(0,\sigma_2^2)\) are independent. Then \(\mathbb{V}(Z) = \sigma_1^2 + \sigma_2^2,\) so \(Z \sim \mathscr{N}_\mathbb{C}(0,\sigma_1^2 + \sigma_2^2)\).

We pay special attention to the equi-variance case \(\sigma_1 = \sigma_2 = \sigma\), where \(Z \sim \mathscr{N}_\mathbb{C}(0,2\sigma^2)\). In particular, when \(2\sigma^2 = 1\), \(Z\) is called a standard complex Gaussian random variable; see [34].

Let \(Z = X + {\rm i}Y\), where \(X\) and \(Y\) are independent centered Gaussians with variance \(\sigma^2\), i.e., \(X,Y \sim \mathscr{N}_\mathbb{R}(0,\sigma^2)\). After normalization, \(X/\sigma, Y/\sigma \sim \mathscr{N}_\mathbb{R}(0,1)\), so \[\sigma^{-2}|Z|^2 = \sigma^{-2}(X^2 + Y^2) = \left(\frac{X}{\sigma}\right)^2 + \left(\frac{Y}{\sigma}\right)^2 \sim \chi_2^2 = \mathscr{E}\left(\frac{1}{2}\right),\] both having PDF \[f_{\sigma^{-2}|Z|^2}(x) = \frac{1}{2} e^{-x/2} \boldsymbol{1}_{[0,\infty)}(x).\]

Furthermore (see 5.2.2), \(|Z|^2\) follows the exponential distribution with rate parameter \(\frac{1}{2\sigma^2}\), that is, \[\label{square} |Z|^2 \sim \mathscr{E}\left(\frac{1}{2\sigma^2}\right).\tag{46}\]

Similarly, for the distribution of \(|Z|\), we have \[f_{|Z|}(x) = \frac{x}{\sigma^2} e^{-x^2/(2\sigma^2)} \boldsymbol{1}_{[0,\infty)}(x).\] This is the PDF of the so-called Rayleigh distribution with parameter \(\sigma\), denoted \(|Z| \sim \mathscr{R}(\sigma)\); see [35].

Let \(Z = R e^{{\rm i} \Theta}\). From the preceding discussion, the radius \(R\) follows the Rayleigh distribution with parameter \(\sigma\), i.e., \(R \sim \mathscr{R}(\sigma)\), and \(R^2\) follows the exponential distribution with rate parameter \(\frac{1}{2\sigma^2}\), i.e., \[R^2 \sim \mathscr{E}\left(\frac{1}{2\sigma^2}\right).\] Furthermore, the phase \(\Theta\) is uniformly distributed on \([0,2\pi)\), i.e., \(\Theta \sim \mathscr{U}[0,2\pi)\); see [1].

5.6 Chernoff bound↩︎

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