  We report a systematic measurement of cumulants, $C_{n}$, for net-proton,
proton and antiproton multiplicity distributions, and correlation functions,
$\kappa_n$, for proton and antiproton multiplicity distributions up to the
fourth order in Au+Au collisions at $\sqrt{s_{\mathrm {NN}}}$ = 7.7, 11.5,
14.5, 19.6, 27, 39, 54.4, 62.4 and 200 GeV. The $C_{n}$ and $\kappa_n$ are
presented as a function of collision energy, centrality and kinematic
acceptance in rapidity, $y$, and transverse momentum, $p_{T}$. The data were
taken during the first phase of the Beam Energy Scan (BES) program (2010 --
2017) at the BNL Relativistic Heavy Ion Collider (RHIC) facility. The
measurements are carried out at midrapidity ($|y| <$ 0.5) and transverse
momentum 0.4 $<$ $p_{\rm T}$ $<$ 2.0 GeV/$c$, using the STAR detector at RHIC.
We observe a non-monotonic energy dependence ($\sqrt{s_{\mathrm {NN}}}$ = 7.7
-- 62.4 GeV) of the net-proton $C_{4}$/$C_{2}$ with the significance of
3.1$\sigma$ for the 0-5\% central Au+Au collisions. This is consistent with the
expectations of critical fluctuations in a QCD-inspired model. Thermal and
transport model calculations show a monotonic variation with $\sqrt{s_{\mathrm
{NN}}}$. For the multiparticle correlation functions, we observe significant
negative values for a two-particle correlation function, $\kappa_2$, of protons
and antiprotons, which are mainly due to the effects of baryon number
conservation. Furthermore, it is found that the four-particle correlation
function, $\kappa_4$, of protons plays a role in determining the energy
dependence of proton $C_4/C_1$ below 19.6 GeV, which cannot be understood by
the effect of baryon number conservation.
