Tropicalization of Haar Measures

• Tropicalization of Haar measures is a framework that transforms translation-invariant measures on non-Archimedean lattices into combinatorial polyhedral semimodules.
• It uses coordinate-wise valuation pushforwards and entropy polynomials to convert classical measure-theoretic data into discrete tropical geometric structures.
• The process reveals an intrinsic tropical linear algebra structure, where canonical generators and uniform exponential distributions emerge in the tropical limit.

The tropicalization of Haar measures on lattices over non-Archimedean local fields formalizes the degeneration of additive, translation-invariant measure theory into piecewise-linear combinatorics on polyhedral semimodules in $\mathbb{R}^n$. A Haar measure on a lattice $L \subset K^n$, normalized to assign probability one to $L$, is pushed forward under coordinate-wise valuation to a discrete measure on $\mathbb{Z}^n$, whose support and distribution are governed by a "tropical" multilinear polynomial derived from the lattice's structure. In the tropical limit—passing to $q \to \infty$ or $t \to 0$—the measure localizes onto the support of a polyhedral complex, which recovers the tropical linear space associated to $L$ and encodes a canonical semimodule structure. This procedure reveals the precise way in which non-Archimedean measure-theoretic data passes to tropical geometric and combinatorial invariants, with critical implications for tropical linear algebra and the study of amoebas and entropy polynomials (Maazouz, 13 Dec 2025).

1. Non-Archimedean Lattices and Haar Measures

Let $K$ denote a non-Archimedean local field, such as $F_q((t))$ or its algebraic closure, with valuation

$$\operatorname{val}: K \to \mathbb{Z} \cup \{\infty\}, \quad \operatorname{val}(0) = \infty,$$

and absolute value $|x| = q^{- \operatorname{val}(x)}$ inducing the $t$-adic topology. Any lattice $L \subset K^n$ is defined as a finitely generated $\mathcal{O}_K$-submodule of full rank $r \leq n$, typically constructed from a full-rank $r \times n$ matrix $A \in \operatorname{Mat}_{r \times n}(K)$ via $L = \mathcal{O}_K \cdot \operatorname{rowspan}(A)$.

Given the local compactness of $K^n$, there exists a unique Haar measure $\mu_{K^n}$ normalized so that $\mu(\mathcal{O}_K^n) = 1$. Restricting and renormalizing yields a probability Haar measure $\mu_L$ on $L$.

2. Pushforward Under Valuation and the Entropy Polynomial

The essential mechanism of tropicalization is the pushforward of the probabilistic Haar measure $\mu_L$ by the coordinate-wise valuation map

$$\operatorname{val}: K^n \to \mathbb{Z}^n, \quad (x_1, \ldots, x_n) \mapsto (\operatorname{val}(x_1), \ldots, \operatorname{val}(x_n)).$$

The resulting discrete probability measure $\nu_L$ on $\mathbb{Z}^n$ admits an explicit survival function in terms of the entropy vector $h(L)$. For every subset $J \subset \{1, \ldots, n\}$ of cardinality $|J| \leq r$, define

$$h_J(L) := \min_{I \subset \{1, \ldots, r\}, |I| = |J|} \operatorname{val} \det A_{I \times J} \in \mathbb{Z} \cup \{\infty\},$$

where $A_{I \times J}$ denotes the minor indexed by $I, J$.

The entropy polynomial $\varphi_L$ (also called the tropical polynomial) in $v = (v_1, \ldots, v_n) \in \mathbb{R}^n$ is given by

$$\varphi_L(v) = \max_{J \subset [n]} \left\{ v_J - h_J(L) \right\},$$

with $v_J = \sum_{j \in J} v_j$, $v_\varnothing = 0$, $h_\varnothing = 0$, and $h_J = \infty$ when $|J| > r$.

A key theorem ([El Maazouz '22]) asserts:

$$\nu_L(\{ w \in \mathbb{Z}^n : w \geq v \}) = \mu_L(L \cap t^v \mathcal{O}_K^n) = q^{-\varphi_L(v)}$$

and for any coordinate $j$,

$$\nu_L(\{ v \}) = q^{-\varphi_L(v)} - q^{-\varphi_L(v + e_j)}.$$

Thus, the survival function of $\nu_L$ is a pure power of $q$, with the exponent given by $\varphi_L$.

3. Polyhedral Complexes and the Tropicalization Process

Transitioning from the discrete measure $\nu_L$ to a tropical geometric object proceeds in two stages.

(a) Support and Semimodule Structure

An integer $v \in \mathbb{Z}^n$ is in the support of $\nu_L$—i.e., $\nu_L(\{ v \}) > 0$—if and only if for all $j = 1, \ldots, n$,

$\varphi_L(v + e_j) - \varphi_L(v) = 1,$

equivalently, $v$ belongs to the tropical semimodule $|\Sigma_L| \cap \mathbb{Z}^n$. The set

$$\operatorname{trop}(L) = \{ \operatorname{val}(x) : x \in L \cap (K^\times)^n \}$$

thus coincides with the integer points of this semimodule.

(b) Cell Decomposition by the Entropy Polynomial

Regarded as a convex, piecewise-linear function, $\varphi_L$ subdivides $\mathbb{R}^n$ into maximal domains of linearity (the fan $K_L$), each labeled by the subset $J \subset [n]$ attaining the maximum in (1). The polyhedral complex $\Sigma_L$ is defined as those cells $\sigma \in K_L$ for which the union of all maximal cell labels equals $[n]$. The support $|\Sigma_L| \subset \mathbb{R}^n$ forms an unbounded, pure $r$-dimensional complex.

A core result (El Maazouz '24, Theorem A) establishes:

1. $$\operatorname{trop}(L) = |\Sigma_L| \cap \mathbb{Q}^n$$; in particular, $$\operatorname{trop}(L \cap (K^\times)^n) = |\Sigma_L| \cap \mathbb{Z}^n$$.
2. The projection of $|\Sigma_L|$ onto $\mathbb{R}^n/\mathbb{R} \cdot (1, \ldots, 1)$ yields the classical tropical linear space $\operatorname{trop}(V)$, with $V = L \otimes_{\mathcal{O}_K} K$.

4. Tropical Semimodule Structure and Canonical Generators

The support $|\Sigma_L|$ admits a canonical tropical linear algebra structure over the semiring $\mathbb{R}_{\geq 0}$. For $u, v \in |\Sigma_L|$,

• the coordinatewise minimum $u \oplus v$ lies in $|\Sigma_L|$,
• for $\lambda \geq 0$, the tropical scalar multiplication $$\lambda \odot u := u + \lambda \cdot (1, \ldots, 1)$$ also lies in $|\Sigma_L|$.

Canonical generators $u_J \in \mathbb{R}^n$ are indexed by $J \subset [n]$, $J \neq \emptyset$, $h_J < \infty$, with

$$(u_J)_j = \begin{cases} h_{J \cup \{j\}} - h_J &\text{if } j \notin J, \ +\infty &\text{if } j \in J. \end{cases}$$

A further result (El Maazouz '24, Theorem B) affirms:

$$|\Sigma_L| = \operatorname{TropCone}_{\mathbb{R}_{\geq 0}}(\{ u_J : J \subset [n], h_J < \infty \}),$$

with every point $x \in |\Sigma_L|$ a tropical linear combination $x = \bigoplus_J \lambda_J \odot u_J$, $\lambda_J \geq 0$.

5. Underlying Proofs and Limit Arguments

The correspondence between Haar measures and entropy polynomials is extracted by covering $V = L \otimes K$ with translates of "box" lattices $t^v \mathcal{O}_K^n$ and applying a non-Archimedean analog of the Cauchy–Binet/determinant formula to count lattice points, with normalization by $\mu_V(L)$. This narrows the exponent in $q^{-(v_J - h_J)}$ down to the minimal valuation of each minor.

Passing to the tropical limit utilizes the Puiseux field $K = \mathbb{C}\{\{ t^{1/N} \}\}$ and sends $t \to 0$, transferring the $q$-adic count to real geometry. Through an infinite-prime strategy and Hilbert–Nullstellensatz, the piecewise-linear (tropical) conditions are “lifted” back to algebraic properties at finite $t$. The amoeba $\operatorname{Log}_{|t|}(A_t)$ of a complex lattice is shown to converge in Hausdorff distance to $|\Sigma_L|$.

The semimodule structure's closure under tropical addition and scaling follows from the $\mathcal{O}_K$-module structure and the ultrametric inequality. Canonical generators suffice for $|\Sigma_L|$ by the super-modularity of $h$ and the explicit definition of $\Sigma_L$.

6. Characteristic Examples and Distributional Behavior

Two instructive examples illustrate the principles:

Example	Matrix $A$	Tropical Polynomial $\varphi_L$	Fan $|\Sigma_L|$ / Amoeba Limit
Rank-2 standard cusp	$$\begin{bmatrix} 1 & 1 & 1 \ 0 & t & t^2 \end{bmatrix}$$	$\max(0, x, y, z, x + y - 1, y + z - 1, x + z - 2)$	2-dimensional fan, projects to tropical 2-plane
Amoeba $\to$ Tropical	$$\begin{bmatrix} 1 & 1 \ 0 & t \end{bmatrix}$$	Wedge: $$\{ (x, y) : 0 \leq y \leq x \} \cup \{ (x, y) : 0 \leq x \leq y \}$$	Amoeba converges to $|\Sigma_L|$

In both cases, the pushforward Haar measure induces a uniform exponential–geometric ("Gauss–Laplace") distribution on the tropical side, with density at $v$ proportional to $e^{-\alpha \varphi_L(v)}$.

7. Synthesis and Equivalence of Classical and Tropical Regimes

The methodology of tropicalizing Haar measures is encapsulated by:

1. Computation of the entropy vector $h_J(L)$ from valuations of minors,
2. Construction of the tropical (entropy) polynomial $\varphi_L(v) = \max_J (v_J - h_J)$ and the pushforward measure with survival function $q^{-\varphi_L(v)}$,
3. Passage, as $q \to \infty$ or $t \to 0$, to the polyhedral semimodule $|\Sigma_L|$ and recovery of the tropical linear space $$\operatorname{trop}(V) = |\Sigma_L|/\mathbb{R} \cdot (1, \ldots, 1)$$.

The equivalence between the original and tropicalized regimes manifests in the formula

$$\mathbb{P}(\operatorname{val}(x) \geq v) = q^{-\varphi_L(v)},$$

with the tropical support $|\Sigma_L|$ characterized by the property that all coordinate "jumps" in $\varphi_L$ are equal to $1$.

The tropicalization of Haar measures thus provides a precise and computable bridge between non-Archimedean probabilistic objects and piecewise-linear tropical structures, with combinatorial, geometric, and measure-theoretic data correspondences encoded explicitly by the entropy polynomial and its induced polyhedral semimodule (Maazouz, 13 Dec 2025).