documentclass[tikz]{standalone}
usepackage{pgfplots}

definecolor{RoyalAzure}{rgb}{0.0, 0.22, 0.66}

newcommandrandompath[2]{%
pgfmathsetseed{#1}%
addplot[#2,domain=0:1,samples=5,smooth] {0.02*(x+0.2)^(-0.4) -0.19 + (x+0.2)+rand*(x+0.32)*(1-(x+0.2))};%
}


pgfplotsset{
/pgfplots/layers/mylayer/.define layer set=
{axis background,axis grid,main,axis ticks,axis lines,axis tick labels,axis descriptions,axis foreground}
{/pgfplots/layers/standard}}

begin{document}

begin{tikzpicture}
    pgfplotsset{set layers=mylayer}%
    pgfplotsset{/dummy/workaround/.style={/pgfplots/axis on top}}

    begin{axis}[width=columnwidth,
        height=0.618columnwidth,
        axis x line=bottom,
        axis y line=left,
        axis line style={line width=1pt},
        /dummy/workaround,
        xlabel style = {font=Large, xshift=31ex,yshift=2.5ex},
        ylabel style = {font=Large, rotate = -90, xshift=3.5ex, yshift=15.5ex},
        xlabel={$theta$},
        ylabel={$langle Mrangle$},
        xmin=0,
        xmax=1,
        ymin=0,
        ymax=1,
        ticks=none,
        ]
    randompath{19.2}{RoyalAzure!10!white, line width=14pt}  % ERROR FUNCTION
    randompath{19.2}{RoyalAzure, line width=1pt}  % ACTUAL FUNCTION
    end{axis}
end{tikzpicture}
end{document}