—   number of variables in each equation ($k\geq3$)

$\xi_k$ is the positive solution of $k=\frac{\xi(1-e^{-\xi})}{1-e^{-\xi}-\xi e^{-\xi}}$.

$\alpha_k=\frac{\xi_k}{k(1-e^{-\xi_k})^{k-1}}$ for $k\geq3$ [1].

Equivalently, with $q$ the largest solution of $q=1-e^{-k\alpha q^{k-1}}$ and $x=k\alpha q^{k-1}$, the threshold is the value of $\alpha$ for which $\alpha q^k=1-e^{-x}-xe^{-x}$ [1].

The density is $\alpha=m/n$, equations per variable; in the finite model $m=\lfloor\alpha n\rfloor$. The average degree of the underlying $k$-uniform constraint hypergraph is $k\alpha$. Each equation uses $k$ distinct variables and a uniform right-hand side. In cuckoo hashing [4] with $k$ hash functions and buckets that hold one key each, $\alpha_k$ is the threshold for the load, the number of keys per bucket [2]. Allowing repeated variable choices gives the same threshold [1].

There is no $k=2$ row. In random $2$-XORSAT the limiting probability of satisfiability is strictly between $0$ and $1$ below density $\alpha=1/2$, average degree $1$, and is $0$ above it, so no density has the property that defines $\alpha_k$ [1].

Peeling variables that occur in at most one equation leaves the $2$-core of the constraint hypergraph, and satisfiability is decided at the density where that core has asymptotically equal numbers of equations and variables [1]. The core thresholds of the Erdős-Rényi random graph use the same Poisson-tail method for graph cores.

Python

from mpmath import mp
mp.dps = 50
k = 3
psi = lambda x: x*(1-mp.e**(-x))/(1-mp.e**(-x)-x*mp.e**(-x))
xi = mp.findroot(lambda x: psi(x) - k, k)
print(xi/(k*(1-mp.e**(-xi))**(k-1)))

B. Pittel and G. B. Sorkin, The satisfiability threshold for $k$-XORSAT, Combinatorics, Probability and Computing 25 (2016), 236-268. (arXiv) (doi)

M. Dietzfelbinger, A. Goerdt, M. Mitzenmacher, A. Montanari, R. Pagh and M. Rink, Tight thresholds for cuckoo hashing via XORSAT, Proceedings of ICALP 2010, Lecture Notes in Computer Science 6198, 213-225. (arXiv) (doi)

Wikipedia: XOR-SAT

Wikipedia: Cuckoo hashing

$k$-core thresholds of the Erdős-Rényi random graph —   the same Poisson-tail calculation for cores of a random graph; this table applies it to the $2$-core of a random $k$-uniform hypergraph

Entries are of type: real number

How well the digits are known: proven

Sources of data: [1], [2]

Table is complete: no (it holds $\alpha_k$ for every $k$ with $3\leq k\leq12$, matching the companion graph-core table; [2] tabulates $3\leq k\leq7$, and from $k=8$ onward the thresholds are within $4\cdot10^{-4}$ of $1$)

How they were obtained:

Each entry is computed in ball arithmetic with 64 guard bits beyond the 100 digits written. The root $\xi_k$ is enclosed by bisection on the sign of $\psi(x)-k$, where $\psi(x)=x(1-e^{-x})/(1-e^{-x}-xe^{-x})$, down to a bracket of half-width $10^{-106}$, with the signs at the two ends checked in ball arithmetic; the value of $x/(k(1-e^{-x})^{k-1})$ on that bracket is the stored ball.

more

Before any entry was written, the values for $3\leq k\leq7$ were compared with the ten-digit table of Dietzfelbinger, Goerdt, Mitzenmacher, Montanari, Pagh and Rink, and every value for $3\leq k\leq12$ was compared with the independent $2$-core characterisation in (3).